Maximum Independent Set

The Antibandwidth and Cyclic Antibandwidth problems are NP-hard graph labeling problems that aim to maximize the minimum (cyclic) distance between labels assigned to adjacent vertices. Extensive research on these problems has resulted in a variety of mathematical formulations and computational approaches. However, their minimum span perspective, in which a prescribed minimum (cyclic) distance is fixed and the objective is to minimize the label span, has received comparatively little attention. In this paper, we consider this complementary perspective by introducing the Minimum Span Antibandwidth/Cyclic Antibandwidth Labeling (MSABL/MSCABL) problems and developing a unified Boolean Satisfiability (SAT)-based framework for solving them. The SAT-based framework formulates MSABL/MSCABL as a sequence of decision problems and exploits their monotonicity to accelerate the search process. We also consider two SAT solving strategies, parallel and incremental SAT solving: the former examines multiple candidate spans concurrently, while the latter reuses a single SAT instance while progressively restricting the label domain. The proposed approaches are evaluated on benchmark instances from the Harwell-Boeing Sparse Matrix Collection and compared with CPLEXCP, CPLEXMIP, and Gurobi. The results show that SAT-based approaches are highly competitive in solution quality, with the parallel approach performing best overall for MSCABL and the incremental approach for MSABL. With the no-hole constraint, they remain competitive with CPLEXCP and significantly outperform CPLEXMIP and Gurobi, particularly for MSCABL. These results demonstrate the effectiveness of SAT solving as an exact approach for MSABL and MSCABL.

We study the maximum strong independent set problem in a finite hypergraph: find the largest vertex set that intersects every hyperedge in at most one vertex. This objective arises whenever each observed block is a local incompatibility constraint but transitive closure across overlapping blocks is not justified. A motivating example is multi-band LSH-MinHash deduplication, where each collision bucket gives local evidence, while connected-component contraction can impose spurious global equivalences. The paper develops an incidence-structural toolkit for this problem. We prove exact reductions for dominance, incidence twins, and weight-1 blocks; derive closed-form and low-weight upper bounds; introduce puncturing and covering certificates that sharpen those bounds; and analyze a layered greedy clustering algorithm driven by block weights and residual incidence. The algorithmic analysis includes feasibility, maximality, conditional optimality, a layered witness-matching upper bound, and incidence-local complexity bounds. The results give correctness, termination, fixed-point, and optimality certificates for broad incidence families, together with examples showing when different certificates separate or coincide.

The Graph Coloring Problem (GCP) is NP-hard and DSATUR stands as one of the fastest heuristics for it despite producing colorings that typically use more colors than state-of-the-art coloring algorithms. We propose SSLD (Semidefinite Spectral Learning with DSATUR), which improves DSATUR by preprocessing a first good color class before letting DSATUR complete coloring the rest of the given graph. We obtain this color class from a Semidefinite Programming (SDP), similar to an SDP used to compute the Lovász theta number. To the best of our knowledge, SSLD is the first approach to improve DSATUR by preprocessing through fixed color classes. We evaluate SSLD against DSATUR and against a naive 1-color-class preprocessing algorithm on DIMACS instances, random graphs (Erdős--Rényi, Watts-Strogatz, Barabási--Albert), Frequency Assignment and Job Shop Scheduling instances. SSLD matches or beats DSATUR in almost every case across over 1600 benchmark instances, and out performs the naive GISD baseline, allows us to confirm the value brought by the SDP-guided choice of the first color class. This quality comes at a runtime cost of roughly 195 times slower that DSATUR, but demonstrating that SDP-guided preprocessing of a first color class is a direction for future improvements.

Despite its ubiquity, clustering lacks a universally accepted definition of what is a cluster. Kleinberg's Impossibility Theorem formalizes this difficulty by showing that no flat clustering method can simultaneously satisfy three natural axioms: scale invariance, richness, and consistency. In this paper, we ask whether this impossibility persists when the output is a hierarchy rather than a single partition. We show that, in contrast to the flat clustering setting, the hierarchical analog of these axioms are jointly satisfiable. In fact, there exist uncountably many hierarchical clustering methods satisfying these axioms, which we call admissible. We explicitly construct several admissible methods, including methods based on well-separated clusters and a non-binary version of single linkage. For certain pairs of admissible methods, the hierarchy produced by one always refines that produced by the other. This refinement relation defines a partial order on the class of admissible methods. This partially ordered set has no greatest element and contains uncountably many pairwise incompatible maximal elements, revealing substantial diversity among admissible methods. Nevertheless, this diversity is constrained: every admissible method contains a hierarchy of sufficiently well-separated clusters, and every finite collection of admissible methods shares such a nontrivial common backbone.

This paper introduces an evolutionary computation approach for consensus in structural Bayesian Network (BN) fusion under the constraint of limited treewidth. The consensus BN aims to reconcile multiple input BNs into a single one that retains key structural features present in the original networks. Treewidth, a graph-based parameter associated with computationally tractable inference, is utilized to restrict the complexity of the resulting network. A genetic algorithm is proposed to look for a BN that codifies as much information about the unrestricted fusion as possible while ensuring the treewidth restriction. Experimental evaluation demonstrates the genetic algorithm's ability to obtain consensus BNs with limited treewidth, providing a valuable tool for aggregating information from diverse sources while returning a computationally actionable model.

The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weighted sum of the cut edges. Over the last years, we engineered a range of fast algorithms for this problem. Our fastest exact algorithm uses an inexact algorithm to obtain a better bound for the problem, reductions that depend on this bound, improved data structures and parallel contraction routines. It is available in the open-source package VieCut and, on real-world instances, outperformed the previously fastest solvers by a factor of up to 2.5 sequentially and up to 12.9 when run in parallel. We improve this algorithm using agentic algorithm engineering (AAE), a methodology that we introduce here, in which autonomous large language model agents run the algorithm engineering cycle on an existing code base: they form hypotheses about where running time is lost, implement them, benchmark the result on a fixed instance set and keep or discard the change. Even though we had already tuned our algorithm by hand extensively, the agent finds significant optimizations, in particular on the DIMACS core instances: factors of 1.28 (sequential) and 1.63 (32 threads) on real-world k-cores, and 6.26 and 127 on the DIMACS core instances.

Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ have positive probability. The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$. We prove that, asymptotically almost surely as $n \to \infty$, a random graph from $\mathbf{G}_n$ has a vertex partition into $r-1$ parts of roughly equal size, the density of edges between the parts is close to $1/2$, and for every $\varepsilon > 0$ the density of edges within any part is less than $\varepsilon$. The asymptotic structural properties are independent of the weight $w$ as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.

The exact Shannon capacity is unknown for every odd cycle beyond the five-cycle $C_5$, making odd cycles a central open problem in zero-error information theory. Improving the known lower bounds requires constructing large independent sets in strong powers of these graphs. Recent AI-assisted work has produced a rapid sequence of improvements: building on the construction of Itty et al., Gao developed a recursive product construction for combining structured independent sets, and Buys, Polak, and Zuiddam (BPZ) subsequently strengthened this through a richer recursion framework. We continue this line of AI-assisted exploration and introduce a heterogeneous refinement of these constructions. The central observation is that the usefulness of an intermediate construction depends not only on the size of its current main independent set, but also on the auxiliary structure it carries into subsequent recursion. Consequently, different parts of that auxiliary structure need not use the same independent set, and different occurrences in a recursion need not use the same intermediate representation. We formalize this for Gao's binary product and derive explicit propagation rules showing how heterogeneous choices strengthen the resulting gadget while leaving its current code size unchanged, then extend the principle to the more general BPZ framework, tailoring constructions to the distinct roles they play within the recursion. Applying these refinements to the seven-cycle $C_7$, we obtain an independent set in $C_7^{\boxtimes 500}$ yielding $Θ(C_7)\ge 3.25883262\ldots$, improving the best known lower bound. Beyond the numerical gain, the results illustrate a general principle for recursive zero-error constructions: intermediate structures with the same dimension and current code size can have different downstream value depending on where and how they are used in the recursion.

We study fair allocations of indivisible goods among agents with heterogeneous monotone valuations. As fair we consider the allocations that are envy-free-up-to-any-good (EFX). Finding if EFX alloca- tions always exist, even for agents with additive valuations, is a major open problem in Fair Division. Christodoulou et al. (2023) introduced the (multi-hyper)graph setting, where agents and goods are represented by vertices and edges of a graph, respectively, and only the endpoints of an edge may have non-zero marginal value for it. We show that for hypergraphs with girth at least 4 and agents with general monotone valuations there always exists an EFX allocation and can be constructed in polynomial time. We generalize our approach to also show that multi-hypergraphs with girth (on the simple hypergraph) at least 4 always admit an EFX allocation, as long as there exists a single vertex whose incident edges have multiplicity at most the size of that edge minus 2; our construction in this case needs pseudo-polynomial time.

We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $Ω(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity. This leads to an optimal refutation of the Alon-Saks-Seymour conjecture, as well as an optimal communication lower bound for the Clique versus Independent Set problem, improving the previous results of Balodis, Ben-David, Göös, Jain and Kothari (FOCS 2021, SICOMP 2023) by several doubly logarithmic factors. As further applications of our construction to query complexity and learning theory, we exhibit: (a) a family of Boolean functions that has an optimal quartic separation between certificate complexity and approximate degree, and (b) a sample compression lower bound of $Ω(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.

Selecting a complete 3D object from a reconstructed scene with minimal user effort is essential for practical scene editing and embodied interaction. Existing 3DGS-based methods either retrain the Gaussian representation to embed per-object labels, or build dense multi-view SAM observations, both requiring heavy computation and dense viewpoint coverage that is rarely available in practice. We present GaussianSelector, a training-free framework for interactive 3D object selection from sparse views and sparse scribble guidance. Operating directly on native Gaussian primitives, we coarsen dense Gaussians into geometrically coherent superpoints and construct a continuity-weighted graph using appearance and spatial cues. Sparse user scribbles are lifted into 3D via visibility-aware transmittance coverage, and selection is solved as a global graph-cut energy minimization that propagates sparse evidence to a complete 3D object. This design naturally supports multi-round refinement, where users iteratively correct the selection from additional viewpoints to progressively improve the result. Experiments demonstrate that GaussianSelector achieves competitive selection quality against state-of-the-art multi-view SAM-based methods, while requiring significantly fewer interaction views and substantially lower computational overhead. These properties make it well suited for human-in-the-loop 3D scene editing and 3D asset extraction in real-world deployment scenarios.

The Shannon capacity $Θ(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $α(G^d)^{1/d}$ for any $d$, where $α(G^d)$ is the independence number of the $d$-th strong product of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, $62530$ in $C_{13}^{6}$, and $8076974$ in $C_{15}^{8}$, improving the best known lower bounds for the Shannon capacity of these graphs to $Θ(C_7)\geq 134753^{1/10}>3.258020$, $Θ(C_{11})\geq 21909^{1/6}>5.289773$, $Θ(C_{13})\geq 62530^{1/6}>6.300109$, and $Θ(C_{15})\geq 8076974^{1/8}>7.301399$. We also improve the best known lower bounds on the independence numbers of several individual strong products of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.

In many cases, the outage of one transmission line in a system can be localized by monitoring the power flow of another line, and machine learning methods can be used to distinguish the cases under uncertainty. In this study, we examine the improvements in line outage localization performance achieved by various ensemble classifiers compared to single-model methods. In the case studies, we compared the classification results with measurement data collected at observed transmission lines (OTLs) selected using three algorithms, i.e, greedy maximum coverage problem (MCP), high-eta, and random selection, based on two sensitivity factors, i.e., line outage distribution factors (LODFs) and line outage impact factors (LOIFs). We found that the OTLs selected by the greedy MCP algorithm yielded the highest F1 score and the ensemble classifiers significantly outperformed a base kNN classifier. The extra-trees bagging technique achieved the highest F1 score in many instances. All the findings were statistically significant.

In this work, we study various graph partitioning problems under a general demand model. In each such task, we are given a graph $G=(V,E,c,w)$ with a capacity function $c\colon E\to \mathbb{N}$ and a demand function $w\colon V\times V\to \mathbb{N}$. Our main focus is the problem of finding a cut $(S, \bar{S})$ minimizing the quantity $$ψ_w( S ) = \frac{c( S, \bar{S} )}{w( S, V )\cdot w( \bar{S}, V )}.$$ Here, $c( S, \bar{S} )$ is the cost of edges between $S$ and the complement of $S$, $\bar{S}$, and $w( S, V )=w( S )+w( S, \bar{S} )$ is the sum of the internal demand within $S$, $w( S )$, and the demand between vertices of $S$ and $\bar{S}$, $w( S, \bar{S} )$. We call $ψ_w( S )$ the \emph{generalized conductance} of the cut $(S, \bar{S})$, and the task of minimizing $ψ_w( S )$ the Generalized Conductance Problem. Our main contribution is an algorithm with an $\mathcal{O}(\log n)$-approximation guarantee for this objective. Our result is achieved via a two-way reduction: first to the well-known Generalized $k$-Multicut Problem, and then to a constrained variant of the classic Sparsest-Cut Problem, with an additional upper-bound constraint on the amount of demand that may be cut. Moreover, we show that the above procedure can be used to obtain an $\mathcal{O}(\log n)$-bicriteria approximation for Graph Partitioning with Demands, where the goal is to find a minimum-cost subset of edges $C$ such that for every component $H$ of $G\setminus C$, $w( H )\leq ρ\cdot w( V )$. This, in turn, yields an $\mathcal{O}(\log n)$-approximation for Hierarchical Clustering with Demands, the problem of finding a hierarchy of cuts that partitions the graph into increasingly refined clusters. For multiplicative demand functions, we improve these guarantees to $\mathcal{O}(\sqrt{\log n})$ and for trees we get an $\mathcal{O}(1)$-approximation for all of our objectives.

Multi-agent language-model systems increasingly route local interactions, yet the runtime interaction graph is often treated as an implementation detail. We study convention formation in open-weight LM populations spanning 1.1B-32B parameters with a naming-game protocol. Restricted first-token scores over tokenizer-safe labels let us measure prompt-conditioned score-state distributions, construct state-similarity graphs, and separate sampled-label agreement from latent state-space consensus. Across controlled interventions, in the main open-weight repair grids, retained partner-label evidence is necessary but not sufficient: homophilous threshold-similarity routing deletes cross-basin exposure and amplifies fragmentation, while bridge-seeking routing often repairs fragmentation when memory is available. In a three-seed mixed four-model grid, threshold-similarity produces no final behavioral or state consensus in 189 setting-seed runs, whereas state-component and label-disagreement bridges recover final behavioral consensus in 14/18 retained-memory runs. Across homogeneous model populations, retained history generally shifts fragmented dynamics toward consensus; the clearest case is Qwen2.5-32B, which reaches stable behavioral and final state consensus in all 18 retained-history well-mixed settings, while threshold-similarity reaches neither form of consensus in 189 settings. Robustness over state thresholds, population size, and vocabulary size preserves the qualitative ordering, and early-window graph-energy features provide useful within-grid diagnostics.

Fault trees are a widely used as effective risk models for complex systems, answering the question "what can go wrong?", especially through minimal cut set analysis. We study fault trees from the perspective of Halpern & Pearl's theory of actual causality. This allows us to use fault trees to answer the question "why has it gone wrong?", which is fundamental to failure diagnostics. We give a complete classification of each of the different notions of actual causality in terms of the fault tree's graph structure and logical structure, and show how minimal cut sets give rise to actual causes.

This paper studies learning-augmented online weighted vertex cover with advice and a parameter $λ\in (0,1)$. We consider two graph cases: bipartite graphs and general graphs. In both settings, the online algorithm must maintain a feasible vertex cover under irrevocable decisions. We show that these problems admit the same robustness--consistency tradeoffs as learning-augmented ski rental. For the bipartite graph model, we give a randomized algorithm that is $\frac{1}{1-e^{-λ}}$-robust and $\fracλ{1-e^{-λ}}$-consistent. For the general graph model, we give a deterministic algorithm that is $(1+\frac{1}λ)$-robust and $(1+λ)$-consistent. We prove that the tradeoffs above are optimal in both settings. We also validate the proposed algorithms through experiments on synthetic and real-world datasets.

Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem. Existing approaches focus on finding a single optimal solution. We introduce FOSC-X, a framework for extracting the top-M globally optimal flat clusterings from local, non-horizontal cuts of a hierarchical cluster tree, while optionally enforcing constraints on the number of clusters. This enables automatic identification of multiple high-quality alternative clusterings that capture different aspects of the hierarchical structure. Without constraints, the top-M problem can be solved in polynomial time using dynamic programming, exploiting the property that locally optimal partial candidates within subtrees can be combined to form globally optimal solutions while automatically determining the number of clusters. However, this can lead to solutions with numbers of clusters that are ultimately undesirable -- e.g., too large to be meaningful or practically analysed within a particular application domain. Imposing cluster-count constraints breaks the optimality property underlying the unconstrained dynamic programming approach, since locally optimal partial candidates may no longer combine into feasible globally optimal solutions. FOSC-X addresses this challenge through a dynamic programming strategy that maintains compact sets of feasible candidates using lower and upper feasibility bounds while pruning infeasible or dominated combinations. The resulting method guarantees optimal rankings of the top-M solutions with linear-time complexity in the number of cluster nodes and dataset size, both with and without cluster-count constraints. Experiments show that FOSC-X efficiently reveals alternative clustering structures overlooked by single-solution extraction methods.

The Quantum Approximate Optimization Algorithm (QAOA) is a leading variational algorithm for combinatorial optimization on near term quantum devices. As circuit depth increases, the number of optimization parameters grows, making the search landscape increasingly nonlinear and difficult to optimize. Previous studies have shown that optimal QAOA parameters often lie on a low dimensional manifold that can be approximated using Principal Component Analysis (PCA) at shallow circuit depths. However, the effectiveness of PCA decreases at higher depths because the underlying parameter manifold becomes increasingly nonlinear. In this work, we investigate Kernel Principal Component Analysis (KPCA) with a radial basis function kernel as a nonlinear dimensionality reduction technique for QAOA parameter optimization. The model is trained using 200 graphs from each of 3 graph families, namely Erdos-Renyi, Barabasi-Albert, and Watts-Strogatz, with graph sizes ranging from 7 to 10 nodes. Performance is evaluated on 30 test graphs containing 12 nodes at circuit depths 1, 2, 4, and 8. Experimental results demonstrate that KPCA consistently outperforms PCA at deeper circuit depths across all graph families. At depth 8, KPCA achieves approximation ratios above 0.86, while PCA declines to approximately 0.81 to 0.83. Both methods reduce the number of quantum circuit evaluations by more than 93 percent relative to unrestricted QAOA optimization. These findings suggest that nonlinear kernel methods more effectively capture the structure of the QAOA parameter manifold and provide a practical approach for scaling variational quantum optimization to deeper circuits.

This paper explores the problem of finding the minimum zero-forcing set on undirected graphs and proposes an adapted machine-learning framework to solve the problem. The minimum zero-forcing set problem is a graph coloring problem where the color of an initial set of nodes propagates throughout a network. The set of nodes is zero-forcing if it forces all uncolored nodes to change color under the constraint of the color-change rule. There are several applications to this problem across different domains such as network science, network control, and designing logical circuits. Finding the minimum zero-forcing set is shown to be NP-hard. We propose a reinforcement learning framework, SD-ZFS, that adapts the S2V-DQN architecture to the ZFS problem. We train several models on this adapted framework and analyze the performance across graph datasets that have varying structures. We evaluate how the models trained on the framework generalize, scale, and transfer to different network types. The results demonstrate the effectiveness of the framework when compared against the optimal solution and greedy heuristic. We provide further insight into how the ZFS problem can be solved through machine-learning and the influence of network structure on the problem.

Despite great advances, finding accurate segmentation remains a challenging task, especially in scenarios with cluttered backgrounds, complex intensity variations and topology appearance. Minimal path models have exhibited their strong ability in addressing image segmentation tasks. However, the performance of minimal paths-based segmentation approaches is heavily influenced by model initialization, hence limiting their application scope in practice. In this work, we propose a novel mask proposal voting framework that overcomes the major drawback of classical approaches, allowing robust segmentation even in complicated scenarios. Firstly, we introduce an efficient method for constructing adaptive domain cuts as a constraint for initializing the region-based min-cut evolution, by which diverse and reliable mask proposal candidates can be generated, substantially increasing the possibility of accurately covering the objective region by these proposals. Secondly, we propose a new mask voting scheme to build a voting score map encoding the final segmentation information. In contrast to classical path voting methods, our model allows incorporating priors to assign different importance to each individual mask. As a consequence, the proposed segmentation model is capable of accurately delineating object boundaries under complex scenarios, and is insensitive to initialization. Experiments demonstrate that our method consistently outperforms state-of-the-art minimal path-based approaches in both accuracy and robustness.

Multi-agent systems (MAS) were expected to overcome the limitation of single-agent systems (SAS) through collaboration. However, under typicality conditions on the task's constraint graph and bounded inter-agent communication, we prove that the success probability of a MAS is closely tied to the connectivity of task constraints, where each agent has limited information-processing capacity. Specifically, the success probability decays exponentially with an information bottleneck that emerges from partitioning the task's constraint graph among agents. We define this quantity as the \emph{minimum cut cost} $C_{\min}$ of the potential constraint graph of each task. This information-theoretic bound applies to both open systems with external feedback and closed systems without. We validate our theory on both synthetic experiments and real-world empirical data from SWE-bench submissions. From our framework, effective MAS design should incorporate task-inherent constraints alongside engineering optimization, and when $\Cmin$ is high, practitioners should restructure tasks rather than simply scaling agents or communication.

We analyze the two combinatorial problems of Dominating Set and Vertex Coloring regarding what kind of local optima are present for various instances. For a variety of graph classes each, we determine whether the induced landscapes are unimodal, plateau-unimodal (all optima are just one plateau), equimodal (all local optima are global) or truly multimodal. We do this for two different neighborhood operators, one based on making only a single change and one also allowing swaps (interchanging two parts of the solution).

We show that the Vehicle Routing Problem (VRP) can be reformulated as a Graph Edit Distance (GED) maximization problem. Under a simple edge-deletion cost model, minimizing total route cost is equivalent to maximizing the total weight of edges deleted from the complete instance graph. This formulation models VRP at the edge level, where solutions are defined by selected edges rather than route sequences, enabling structural analyses that are difficult in classical formulations: per-edge attribution of solution quality, decomposition of the optimality gap, characterization of solution sparsity, and identification of edges that are hard to reach by greedy construction. Theoretically, we establish a merge-decomposition theorem showing that Clarke-Wright savings equal per-merge GED increments, and an approximation-transfer theorem that turns GED approximation ratios into VRP cost bounds. Using this reformulation, we analyze 90 CVRP benchmark instances with known optimal solutions. We find that optimal routing graphs use only 5.5% of available edges, that approximately 3.0% of optimal edges are consistently not found by Clarke-Wright heuristics under repeated restarts, and that the cost gap decomposes into missed optimal edges and substituted non-optimal edges of comparable total weight. The edge-additive objective provides a natural per-edge supervision signal for future graph neural network approaches to edge prediction, suggesting a potential connection to graph neural network approaches that we leave for follow-up work.

Evaluating large language models (LLMs) across comprehensive benchmarks is expensive and time-consuming. We propose a graph-based prompt selection framework that models each benchmark as a similarity graph -- nodes are prompts connected if their embedding-space distance falls above a configurable threshold -- and applies Maximum Independent Set (MIS) algorithms to select a maximally diverse, non-redundant subset. We evaluate four MIS solvers (CPLEX, GREEDY, Online-MIS, ReduMIS) across six embedding models, three distance measures, six percentile thresholds, and four benchmarks (GPQA, IFEval, MMLU-Pro, Omni-MATH) covering 66 LLMs. Our central hypothesis -- that repeated selection under different random seeds yields consistent LLM rankings that may also differ from the full-benchmark baseline -- is strongly confirmed: Kendall's $W \geq 0.90$ in 99.2% of stochastic configurations (mean $W = 0.997 \pm 0.008$), while at higher percentile thresholds selected subsets achieve 25--48% prompt reduction on average. Ranking divergence from the full benchmark ($ρ< 0.95$) occurs in only 15.95% of configurations, concentrated at low thresholds ($p_{10}$--$p_{20}$) and benchmarks (GPQA, IFEval), identifying overly dense graphs as the primary failure mode.

Graph neural networks (GNNs) are limited to modeling pairwise interactions, while higher-order models based on cell complexes achieve greater expressivity but often suffer from poor scalability. We introduce simplified and factored cellular Weisfeiler Leman tests (sCWL and fCWL), which preserve the expressivity of the CWL test while improving computational efficiency. We further introduce the maximal clique complex, enabling scalable CWNs with reduced time and memory complexity while retaining strong empirical performance. To avoid explicit clique enumeration, we propose CliqueWalk, a biased random walk that samples maximal cliques and scales linearly with graph size. These contributions yield a scalable topological learning framework for higher-order graph representation.

Baldwinian and Lamarckian evolution have existed for a long time in evolutionary algorithms (EAs) without ever dominating the academic literature or practical applications. In this work, we use modern empirical and theoretical methods to revisit Lamarckian and Baldwinian evolution and rigorously compare them with the generic Darwinian evolution. On the empirical side, we run a comprehensive suite of experiments on graphs from six different datasets from the recent GraphBench benchmark on Maximum Independent Set and Maximum Cut problems. Our results show that Baldwinian and Lamarckian evolution consistently outperform Darwinian evolution, confirming the great potential of local search augmented evolutionary algorithms. Notably, in the great majority of cases, all EAs outperform recent deep learning baselines and approach the performance of highly specialised heuristic and exact solvers. We furthermore report a high-performing set of generalist parameters for all studied evolution types that we hope will be of use to practitioners in future. On the theoretical side, we extend the existing DeceptiveLeadingBlocks benchmark to arbitrary block length $k$. For all constant $k$, we then prove asymptotically tight runtime bounds for the $(1+1)$ EA in the three evolution types on this benchmark. For Baldwinian evolution, these are independent of $k$, whereas for the other two evolution types, the runtimes steeply increase with growing value of $k$.

The Maximal Covering Location-Interdiction Problem (MCLIP) is a classic bi-level optimization problem, which is fundamental to resilient infrastructure planning yet remains computationally intractable. Specifically, the upper level determines facility locations to maximize coverage, while the lower level executes worst-case interdiction to minimize the coverage. The strong coupling between the upper and lower levels, combined with their respective high combinatorial complexity, renders traditional methods ineffective. To bridge this gap, we propose a Dual-Agent Deep Reinforcement Learning (DADRL) framework based on adversarial learning, comprising a location agent corresponding to the upper level and an interdiction agent corresponding to the lower level. Our contributions are threefold: (1) The location agent is trained simultaneously against an evolving interdiction agent, making it effectively capture the dynamic competitive interplay between the upper and lower levels; (2) To fully exploit the learned capabilities of the interdiction agent, we propose a Surrogate-based Ensemble Inference Strategy that utilizes the trained interdiction agent as a high-fidelity surrogate to guide the decisions of location agent; (3) Extensive experiments on synthetic and real-world datasets demonstrate that our approach achieves superior computational efficiency while maintaining highly competitive solution quality compared to other baselines. Furthermore, our DADRL framework is model-agnostic to network structures, while its underlying adversarial learning paradigm demonstrates strong potential for solving other bi-level optimization problems.

The communication complexity of a voting rule is the worst-case number of bits that n voters must transmit to a central authority under the most efficient elicitation protocol in an election with m candidates. We study the communication complexity of Instant-Runoff Voting (IRV). Conitzer and Sandholm [2005] established an upper bound of O(n (log m)${}^2$), but did not provide a matching lower bound beyond $Ω$(n log m). We resolve this open problem by raising the lower bound to $Ω$(n (log m)${}^2$) using the fooling set technique, thereby showing that the communication complexity of IRV is $Θ$(n (log m)${}^2$). We further show that this complexity drops to $Θ$(n log m) under the single-peakedness restriction, and that both the IRV-Average variant and Single Transferable Vote (STV), the multiwinner extension of IRV, have the same asymptotic communication complexity as IRV.

We study fixed-cardinality maximization of the inverse-matrix Solow--Polasky diversity, equivalently finite metric magnitude for the exponential kernel, on one-dimensional and ordered metric sets. The analysis starts from the known finite-line gap formula for the exponential kernel, which writes the excess inverse-matrix diversity as a sum of functions of consecutive gaps. Building on this formula, the main interval theorem proves that, for every $k\geq 2$, the unique maximizing $k$-point subset of $[0,1]$ is the equally spaced set. Thus the objective selects a uniform gap representation on the real line. A converse kernel proposition shows that, among normalized non-increasing distance kernels, requiring the corresponding adjacent-gap additive structure forces the exponential family. Further results transfer the interval theorem to ordered $\ell_1$ (L1, or Manhattan) curves by isometry: the maximizing sets are uniform in accumulated $\ell_1$ length. As a consequence, monotone biobjective Pareto fronts admit Solow--Polasky optimal finite approximations that are uniformly spaced in accumulated objective-space change, a natural representation when all parts of a continuous front should be covered. Examples, including a dense connected front and a finite disconnected ZDT3 front, illustrate how the continuous uniform-gap result appears on discrete candidate sets. Solow-Polasky diversity; diversity measures; finite metric magnitude; L1 distance; uniform spacing; Pareto-front approximation; multiobjective optimization; fixed-cardinality subset selection

Data selection seeks to identify a compact yet informative subset from large-scale training corpora, balancing sample quality against collection diversity. We formulate this problem as a Weighted Independent Set (WIS) on a similarity graph, where nodes represent data samples weighted by influence, and edges connect semantically redundant pairs. This formulation naturally yields subsets that are simultaneously high-quality and diverse. However, two challenges arise in practice: naive node weights fail to distinguish informative signals from gradient noise, and edge construction under heterogeneous domain distributions produces structurally imbalanced graphs that bias selection toward sparse regions. To address these issues, we introduce two principled refinements from a unified graph perspective: (1) \textit{node value calibration} that restricts influence estimation to the bilateral salient subspace to ground node importance in task-relevant signals rather than surface-level statistics; (2) \textit{local scale normalization} that adapts edge thresholds to local neighborhood density, mitigating graph imbalance induced by cross-domain distribution shifts. Together, these components yield a robust and scalable data selection pipeline dubbed SEED. We further construct \texttt{Honeybee-Remake-SEED-200K}, a compact multimodal dataset curated by SEED. Extensive experiments show that SEED consistently outperforms state-of-the-art methods on instruction tuning, visual instruction tuning, and semantic segmentation across diverse model families.

The multicut problem is an NP-hard combinatorial optimization problem with diverse applications in fields such as bioinformatics, data mining and computer vision. Graph neural networks have been defined for the multicut problem but can be adapted further to its specific objective function and constraints. In this article, we introduce such an adapted graph neural network architecture in which features are assigned only to edges, and the computation of messages is based on triangles in the underlying graph. Experiments with synthetic and real-world instances with up to 200 nodes show that our method outperforms state-of-the-art heuristic solvers in terms of solution quality while maintaining feasible runtimes. For some instances, our method finds optimal solutions in seconds whereas exact solvers need hours to find and certify optimal solutions.

Computing a minimum $s$-$t$ cut in a graph is a solution to a wide range of computer vision problems, and is often done using the Boykov-Kolmogorov (BK) algorithm. In this paper, we revisit the BK algorithm from both a theoretical and practical point of view. We improve the analysis of the time complexity of the BK algorithm to $O(mn|C|)$ and propose a new algorithm, the fast and compact BK (fcBK) algorithm, with a time complexity of $O(m|C|)$, where $m$, $n$, and $|C|$ are the number of edges, number of vertices, and the capacity of the cut, respectively. We additionally propose a compact graph representation that allows our implementation to find a minimum $s$-$t$ cut in a graph with upwards of $10^9$ vertices and $10^{10}$ edges on a machine with 128 GB of memory. We find our implementation of the BK algorithm to be the fastest available implementation of the BK algorithm when evaluating on a comprehensive set of benchmark datasets, highlighting the importance of memory-efficient implementations. We make our implementations publicly available for further research and implementation development within minimum $s$-$t$ cut algorithms.

We study a graph pathfinding problem Distance-$r$ Independent Unlabeled Multi-Agent Pathfinding, finding a set of collision-free paths between two sets where agents must stay at pairwise distance at least $r+1$ at all times. This additional constraint, generalizing collision modeling for classical MAPF, targets aspects of real-world multi-agent coordination. This additional distance constraint makes feasibility (i.e., whether a solution exists) PSPACE-complete, in contrast to standard (unlabeled) MAPF, where it can be decided in polynomial time. We address the challenge via two complementary approaches: (i) reduction-based optimal algorithms with a feasibility-preserving compression procedure, and (ii) a configuration generator-based search. Despite the hardness, empirical results show that our algorithm can handle hundreds of agents in a practical timeframe.

Treating and preventing Human Immunodeficiency Virus (HIV) remains a critical global health challenge. While antiretroviral therapy provides a path toward viral suppression -- effectively eliminating an individual's transmission risk -- systemic resource constraints limit the reach of intervention efforts. This work addresses the strategic distribution of intensive resources among virally unsuppressed individuals to minimize the expected cascade of new infections within a transmission network. We formalize this challenge as a novel constrained optimization problem where we have resources to "treat" $k$ out of a set $\mathbf{P}$ of virally unsuppressed individuals, and establish its theoretical connections to existing computational literature. We then propose Cascade-Aware Suppression of Transmission (CAST), a polynomial-time $(δ, ε)$-approximation algorithm that achieves a $2\sqrt{|\mathbf{P}|}$ approximation ratio by leveraging connections to the Minimum-$k$-Union (MkU) problem and Hoeffding-style concentration bounds. Extensive evaluations on real-world HIV networks demonstrate that CAST outperforms standard public health and computer science baselines. Furthermore, we show that CAST is empirically robust across diverse infectious disease networks, varied edge probability initializations, and settings involving imperfect network data.

We introduce a use of the $M$-cover (or $M$-layer) transform for machine learning. The method replicates a model $M$ times, but instead of coupling the copies through parameter averaging or an explicit attractive force, as in replicated SGD or Elastic SGD, it rewires the contexts in which local learning messages are computed. Each local loss is evaluated on a routed model whose parameters are drawn from different copies according to permutations sampled from a structured mixing kernel $Q$. Training then uses the original local update rule, while the resulting learning messages are redistributed across the copies through these routed computational paths. Thus $Q$ defines a topology for message transport and controls the long-loop structure of the lifted factor graph. We formulate this construction for perceptrons, committee machines, and multilayer perceptrons, showing that the same principle applies from discrete models to differentiable neural networks. The resulting framework provides a mechanism for improving generalization through structured message sharing rather than replica collapse or parameter-space coupling.

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity $\mathcal{O}(n^2 + ne)$, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to $10.6 \times$ compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.

We present an accelerated relax-and-round algorithm for concave coverage problems, which generalize the classic maximum coverage problem. Building on the relax-and-round framework of Barman et al. [STACS 2021], we propose two significant improvements. First, we replace the linear programming (LP) relaxation step with a projected accelerated gradient method applied to a smooth surrogate objective to achieve a $\widetilde{O}(mn \varepsilon^{-1})$ running time. Second, we use a specialized rounding scheme for the hypersimplex that combines the Carathéodory decomposition algorithm in Karalias et al. [NeurIPS 2025] with randomized swap rounding of Chekuri et al. [FOCS 2010]. We prove tight approximation ratios for new reward functions, including a $0.827$-approximation for the logarithmic reward $\varphi(x) = \log(1 + x)$. Finally, we conduct maximum multi-coverage experiments on synthetic and real-world graphs, demonstrating that our algorithm outperforms approaches that use state-of-the-art LP solvers.

We introduce Graph Normalization (GN), a principled dynamical system on graphs that serves as a differentiable approximation engine for the NP-hard Maximum Weight Independent Set (MWIS) problem. MWIS encompasses many combinatorial challenges, including optimal assignment, scheduling, set packing, and MAP inference in discrete Markov Random Fields. Unlike Belief Propagation, we prove GN always converges to a binary indicator of a Maximum Independent Set. GN realizes a fast quasi-Newton descent through an exact Majorization-Minimization step, systematically improving the MWIS relaxed primal objective. We establish an equivalence between GN and the Replicator Dynamics of a nonlinear evolutionary game, where vertices compete for inclusion in an independent set. While a non-potential game, the GN game follows Fisher's Fundamental Theorem of Natural Selection, where the average fitness equals the MWIS primal objective and strictly increases. This connection leads to a weighted extension of the Motzkin-Straus theorem, showing MISes are in bijection with the local minima of a quadratic form over a tilted simplex. For the Assignment Problem, GN acts as a variant of the Sinkhorn algorithm that naturally converges to a hard assignment while generalizing to arbitrary constraint graphs. We demonstrate GN's performance as a fast binarization engine for the state-of-the-art Bregman-Sinkhorn relaxed MWIS solver. On real-world benchmarks with up to 1M edges, GN identifies solutions within 1% of the best known results in seconds on a CPU. GN opens new avenues for deep learning architectures requiring differentiable, "hard" decisions under constraints, with applications in structured sparse attention, dynamic network pruning, and Mixture-of-Experts. Beyond core AI, the GN framework enables end-to-end learning of constrained optimization in computer vision, computational biology, and resource allocation.

This paper proposes a novel algorithm for semisupervised learning. This algorithm learns graph cuts that maximize the margin with respect to the labels induced by the harmonic function solution. We motivate the approach, compare it to existing work, and prove a bound on its generalization error. The quality of our solutions is evaluated on a synthetic problem and three UCI ML repository datasets. In most cases, we outperform manifold regularization of support vector machines, which is a state-of-the-art approach to semi-supervised max-margin learning.

Post-Training Quantization (PTQ) compresses large language models to low bit-widths using a small calibration set, and its quality depends strongly on which samples are chosen. We identify a failure mode in which calibration samples fail to activate outlier channels, hidden dimensions with unusually large activations, causing the quantizer to underestimate their dynamic range and producing per-channel reconstruction errors that dominate layer-wise loss. Motivated by this observation, we argue that PTQ calibration quality is governed more by weighted outlier-channel coverage than by generic sample representativeness, and formulate calibration selection as a weighted set cover problem over outlier channels. The objective is monotone submodular, and the greedy algorithm, COVERCAL, operates on pre-computed activation statistics and requires no GPU time at selection. We further show that the weight choice is internally consistent: under a stylized clipping model, missed weighted coverage upper-bounds surrogate loss, justifying the weighted coverage objective as principled rather than purely empirical. Across LLaMA-2, LLaMA-3, and Mistral, under AWQ and GPTQ backends and five downstream evaluations, COVERCAL improves over random, max-perplexity, max-activation-variance, and stratified baselines, with the largest gains at small calibration budgets. At INT4 with 128 samples, COVERCAL improves MMLU by 1.2 to 1.5 points over random calibration and reduces perplexity degradation by 15 to 30%; with 64 samples, it matches or exceeds random calibration at 256. The contribution is not a new PTQ backend but a formulation of calibration selection as weighted outlier coverage, with a simple, efficient algorithm and a surrogate-based justification.

Hierarchical clustering is a fundamental task in data analysis, but classical methods have long lacked a principled objective function. Dasgupta [STOC 2016] took an important step toward addressing this gap by proposing a well-motivated objective function for cluster trees. Cohen-Addad et al. [J. ACM 2019] subsequently introduced the notion of admissibility: an objective function is admissible if, whenever the input similarity matrix admits generating trees, its minimizers are precisely those generating trees. They also gave a necessary and sufficient condition for admissibility within a family of objective functions based on aggregate intercluster similarity. We refer to this family as sum-type objective functions. However, apart from Dasgupta's original objective function, no explicit admissible objective functions in this family were provided. In this paper, we study admissible objective functions for hierarchical clustering in two directions. For sum-type objective functions, we give a complete characterization when the scaling function is a symmetric polynomial of degree at most two, and we derive sufficient conditions for degree-three polynomials. We also show that the recursive sparsest cut algorithm achieves an O$(φ)$-approximation ratio for the admissible objective functions covered by our characterization, where $φ$ is the approximation factor of the sparsest cut subroutine. We then introduce max-type objective functions, where cluster interaction is measured by maximum, rather than aggregate, intercluster similarity. For this class, we characterize which objective functions are admissible for arbitrary symmetric scaling functions and give a complete characterization when the scaling function is a symmetric polynomial of degree at most two.

Identifying critical nodes in complex networks is a fundamental task in graph mining. Yet, methods addressing an all-or-nothing coverage mechanics in a bipartite dependency network, a graph with two types of nodes where edges represent dependency relationships across the two groups only, remain largely unexplored. We formalize the CriticalSet problem: given an arbitrary bipartite graph modeling dependencies of items on contributors, identify the set of k contributors whose removal isolates the largest number of items. We prove that this problem is NP-hard and requires maximizing a supermodular set function, for which standard forward greedy algorithms provide no approximation guarantees. Consequently, we model CriticalSet as a coalitional game, deriving a closed-form centrality, ShapleyCov, based on the Shapley value. This measure can be interpreted as the expected number of items isolated by a contributor's departure. Leveraging these insights, we propose MinCov, a linear-time iterative peeling algorithm that explicitly accounts for connection redundancy, prioritizing contributors who uniquely support many items. Extensive experiments on synthetic and large-scale real datasets, including a Wikipedia graph with over 250 million edges, reveal that MinCov and ShapleyCov significantly outperform traditional baselines. Notably, MinCov achieves near-optimal performance, within 0.02 AUC of a Stochastic Hill Climbing metaheuristic, while remaining several orders of magnitude faster.

Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to generate and formalize correctness proofs in Lean. This case study highlights the potential of integrating automated reasoning, large language models, and formal verification to accelerate mathematical discovery. We argue that such tool-driven workflows will play an increasingly central role in experimental mathematics.

In an election where $n$ voters rank $m$ candidates, a Condorcet winning set is a committee of $k$ candidates such that for any outside candidate, a majority of voters prefer some committee member. Condorcet's paradox shows that some elections admit no Condorcet winning sets with a single candidate (i.e., $k=1$), and the same can be shown for $k=2$. On the other hand, recent work proves that a set of size $k=5$ exists for every election. This leaves an important theoretical gap between the best known lower bound $(k\geq 3)$ and upper bound $(k \leq 5)$ for the number of candidates needed to guarantee existence. We aim to close the gap between the existence guarantees and impossibility results for Condorcet winning sets. We explore an automated reasoning approach to tighten these bounds. We design a mixed-integer linear program (MILP) to search for elections that would serve as counter-examples to conjectured bounds. We employ a number of optimizations, such as symmetry breaking, subsampling, and constraint generation, to enhance the search and model effectively infinite electorates. Furthermore, we analyze the dual of the linear programming relaxation as a path towards obtaining a new upper bound. Despite extensive search on moderate-sized elections, we fail to find any election requiring a committee larger than size 3. Motivated by our experimental results in this direction, we simplify the dual linear program and formulate a conjecture which, if true, implies that a winning set of size 4 always exists. Our automated reasoning results provide strong empirical evidence that the Condorcet dimension of any election may be smaller than currently known upper bounds, at least for small instances. We offer a general-purpose framework for searching elections in ranked voting and a new, concrete analytical path via duality toward proving that smaller committees suffice.

Chromatic Correlation Clustering (CCC) extends Correlation Clustering by assigning semantic colors to edges and requiring each cluster to receive a single color label. Unlike standard CC, whose LP relaxation has integrality gap 2 on complete graphs and admits a 2.06-approximation, the analogous LP for CCC has a strict lower bound of 2.11, and the best known LP-rounding algorithm achieves 2.15. We explain this gap by isolating the source of difficulty: cross-edge chromatic interference. Neutral edges, whose color does not match the candidate cluster color, create an irreducible cost absent from standard CC and force any color-independent rounding scheme to pay an additional mismatch penalty. We make four contributions. First, we prove a Global Integrality Gap Decomposition Theorem showing that the gap of any color-independent CCC rounding algorithm equals the standard CC gap plus an irreducible chromatic penalty Delta(L) > 0. Second, we solve the associated min-max problem and derive the staircase formula Delta(L) = ((L-1)/L) Delta_infinity, where Delta_infinity is approximately 0.0734. In particular, the two-color gap is 2.0967, separating CCC from standard CC already at L = 2. Third, we introduce Color-Coupled Correlation Clustering (C4). Adding the valid global constraint sum_c x_uv^c >= L-1 and a correlated interval-packing rounding scheme makes neutral edges behave like classical negative edges, recovering the optimal 2.06 approximation and bypassing the 2.11 lower bound for the uncoupled LP. Fourth, experiments on extremal instances, real multi-relational networks, and fairness benchmarks validate the theory: empirical LP gaps follow the predicted staircase, and C4 matches the unconstrained approximation ratio under fairness constraints.

Pareto optimization via evolutionary multi-objective algorithms has been shown to efficiently solve constrained monotone submodular functions. Traditionally when solving multiple problems, the algorithm is run for each problem separately. We introduce multitasking formulations of these problems that are an effective way to solve multiple related problems with a single run. In our setting the given problems share a monotone submodular function $f$ but have different knapsack constraints. We examine the case where elements within a constraint have the same cost and show that our multitasking formulations result in small Pareto fronts. This allows the population to share solutions between all problems leading to significant improvements compared to running several classical approaches independently. Using rigorous runtime analysis, we analyze the expected time until the introduced multitasking approaches obtain a $(1-1/e)$-approximation for each of the given problems. Our experimental investigations for the maximum coverage problem give further insight into the dynamics behind how the approach works and doesn't work in practice for problems where elements within a constraint also have varied costs.

Given a weighted undirected graph, a number of clusters $k$, and an exponent $z$, the goal in the $(k, z)$-clustering problem on graphs is to select $k$ vertices as centers that minimize the sum of the distances raised to the power $z$ of each vertex to its closest center. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact $(k, z)$-clustering solution in the induced shortest-path metric. While efficient dynamic $k$-center approximation algorithms on graphs exist [Cruciani et al. SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic $(k,z)$-clustering problem. As the main result of this paper, we develop a randomized incremental $(k, z)$-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of $\tilde O(k m^{1+o(1)}+ k^{1+\frac{1}λ} m)$, where $λ\geq 1$ is an arbitrary fixed constant. Our incremental algorithm consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size $\tilde{O}(k)$ with a total update time of $m^{1+o(1)}$ over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant, a property that may be of independent interest. In the second stage, we maintain a constant-factor approximate $(k,z)$-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static $(k,z)$-clustering algorithm.

A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior. Let $S(n)$ be the minimum number of hyperplanes needed to slice $Q_n$. We prove that $S(n) \leq \lceil \frac{4n}{5} \rceil$, except when $n$ is an odd multiple of $5$, in which case $S(n) \leq \frac{4n}{5} +1$. This improves upon the previously known upper bound of $S(n) \leq \lceil\frac{5n}{6} \rceil$ due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in $Q_n$ that can be sliced using $k<n$ hyperplanes. We prove the improved upper bound on $S(n)$ by constructing $8$ hyperplanes slicing $Q_{10}$ aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.

Neural Combinatorial Optimization (NCO) has mostly focused on learning policies, typically neural networks, that operate on a single candidate solution at a time, either by constructing one from scratch or iteratively improving it. In contrast, decades of work in metaheuristics have shown that maintaining and evolving populations of solutions improves robustness and exploration, and often leads to stronger performance. To close this gap, we study how to make NCO explicitly population-based by learning policies that act on sets of candidate solutions. We first propose a simple taxonomy of population awareness levels and use it to highlight two key design challenges: (i) how to represent a whole population inside a neural network, and (ii) how to learn population dynamics that balance intensification (generating good solutions) and diversification (maintaining variety). We make these ideas concrete with two complementary tools: one that improves existing solutions using information shared across the whole population, and the other generates new candidate solutions that explicitly balance being high-quality with diversity. Experimental results on Maximum Cut and Maximum Independent Set indicate that incorporating population structure is advantageous for learned optimization methods and opens new connections between NCO and classical population-based search.

We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial optimization, e.g., the densest subgraph, maximum clique, and maximum edge biclique problems, and has wide application the study of complex networks. Much recent research has focused on the development of sufficient conditions for exact solution of the densest submatrix problem via convex relaxation. The vast majority of these sufficient conditions establish identification of the densest submatrix within a graph containing exactly one large dense submatrix hidden by noise. The assumptions of these underlying models are not observed in real-world networks, where the data may correspond to a matrix containing many dense submatrices of varying sizes. We extend and generalize these results to the more realistic setting where the input matrix may contain \emph{many} large dense subgraphs. Specifically, we establish sufficient conditions under which we can expect to solve the densest submatrix problem in polynomial time for random input matrices sampled from a generalization of the stochastic block model. Moreover, we also provide sufficient conditions for perfect recovery under a deterministic adversarial. Numerical experiments involving randomly generated problem instances and real-world collaboration and communication networks are used empirically to verify the theoretical phase-transitions to perfect recovery given by these sufficient conditions.

Balancing resource efficiency and fairness is critical in networked systems that support modern learning applications. We introduce the \emph{Fair Minimum Labeling} (FML) problem: the task of designing a minimum-cost temporal edge activation plan that ensures each group of nodes in a network has sufficient access to a designated target set, according to specified coverage requirements. FML captures key trade-offs in systems where edge activations incur resource costs and equitable access is essential, such as distributed data collection, update dissemination in edge-cloud systems, and fair service restoration in critical infrastructure. We first give a structural characterisation of the single-terminal case, showing that it is equivalent to the rooted Covering Steiner problem. We prove that FML is NP-hard and admits no $((1-ε)\ln |\mathcal{C}|)$-approximation for $|\mathcal{C}|$ groups, already on a star, while for any fixed number of groups it inherits a constant-factor approximation and remains APX-hard. We then present probabilistic approximation algorithms for the two-group, single-terminal case: an algorithm whose tree subroutine is exact, hence optimal on tree-structured networks and $\mathcal{O}(\log |V|)$ in expectation on general graphs, together with a faster bicriteria variant whose coverage violation degrades gracefully with the merge depth of the tree computation. For practical scalability, we additionally introduce a graph-native variant based on a shortest-path-tree reduction. Empirical results show that FML enforces group-level fairness, while the graph-native variant substantially improves scalability and achieves competitive activation cost.

Although the Maximum Clique Problem (MCP) has been extensively studied and features a rich ecosystem of exact solvers, empirical evidence shows that solver performance varies substantially across graph families. Consequently, selecting an appropriate algorithm for a given instance remains an open and practically important challenge that has received little systematic attention. We address this gap by developing an instance-aware selection framework that systematically combines global statistical descriptors with learned topological representations. We construct a comprehensive benchmark by evaluating four state-of-the-art exact solvers on a diverse collection of graph instances and deriving both global statistical and local structural features. An evaluation of conventional classifiers establishes Random Forest as a strong baseline and reveals that connectivity and topological features are key predictors of performance. Motivated by these observations, we introduce a dual-channel architecture that jointly leverages a Graph Attention Network for capturing local neighborhood patterns and a Multilayer Perceptron for modeling global statistical features. Extensive experiments show that the proposed dual-channel model consistently surpasses classical baselines and the single-best solver, achieving 90.43% test accuracy. These findings demonstrate the value of integrating local topological encoding with global statistical cues for combinatorial algorithm selection. Code and models are available at: https://anonymous.4open.science/r/GAT-MLP-7E5F.

This manuscript provides a comprehensive review of the Maximum Clique Problem, a computational problem that involves finding subsets of vertices in a graph that are all pairwise adjacent to each other. As such, this review is a continuation of the series of previous reviews from 1994, 1999 and 2014. The manuscript covers in a simple way classical algorithms and includes a review of recent developments in graph neural networks and quantum algorithms.

In this paper, we consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of $2$ sub-gaussian distributions in $\mathbb{R}^p$. We consider semidefinite programming relaxations of an integer quadratic program that is formulated essentially as finding the maximum cut on a graph, where edge weights in the cut represent dissimilarity scores between two nodes based on their $p$ features. We define the signal-to-noise ratio (SNR) as $s^2 := \min\{n p γ^2, Δ^2\}$, where $Δ^2 := p γ$ denotes the $\ell_2^2$ distance between the two cluster centers. Our contributions are twofold. First, we provide a unified framework for analyzing three computationally efficient algorithms: SDP1, BalancedSDP, and Spectral clustering, yielding universal polynomial-rate misclassification guarantees for all three algorithms. Moreover, our theory allows for partial recovery (success rate $< 100\%$) as long as $s^2$ is lower bounded by a constant. Second, we prove that the misclassification errors for SDP1 and BalancedSDP decay exponentially with respect to the SNR $s^2$ and the BalancedSDP requires no explicit debiasing when the two clusters have equal sizes. To our knowledge, this is the first time such results are obtained for semidefinite relaxations of MAX CUT in population clustering. We provide simulation evidence illuminating the theoretical predictions.

Deep neural networks (DNNs) are highly vulnerable to adversarial attacks. Ideally, a robust model should perform well on both perturbed training data and unseen perturbed test data. While DNNs can fit perturbed training data, generalizing to perturbed test data remains a significant challenge. This motivates the study of generalization guarantees from a learning theory perspective. This paper focuses on adversarial Rademacher complexity (ARC), first introduced by Khim and Loh (2018) and Yin et al. (2019). Their work primarily addressed linear functions and highlighted the open question of how to bound ARC for neural networks. Since then, several attempts have been made, with the latest results applying ARC only to two-layer neural networks. The main challenge arises from the dynamic nature and unknown closed-form solution of adversarial examples. In this paper, we resolve this issue and provide the first bound on ARC for deep neural networks. Our bound is qualitatively comparable to Rademacher complexity bounds in similar settings. The key ingredient is a new concept we introduce, termed intermediate adversarial examples, along with a framework for calculating the covering number that is compatible with them. Finally, we present experiments to analyze poor robust generalization, demonstrating that the weight norm is a crucial factor influencing the robust generalization gap.