Constraint Propagation

A feedback-based word deduction framework based on the Jotto problem is proposed, and the problem space is represented as a weighted graph where all valid words correspond to nodes, and the edge weight is defined by the number of common letters between the two words. Finally, the gameplay is defined as an iterative constraint propagation mechanism where feedback is used to iteratively narrow the incompatible space of the graph, facilitating the reduction of the hypothesis space in a structured and interpretable manner. In contrast to existing approaches, where the problem space is typically defined for fixed-length isograms, the proposed framework generalizes to variable-length words (between 3 and 8 letters) and naturally extends to repeated letter cases, facilitating the treatment of realistic Jotto problem instances within a unified framework for the first time. The proposed framework's applicability and solver dynamics are also discussed through an interactive implementation and a qualitative case study, respectively. Significant automated tests on approximately 3,000 simulated gameplay scenarios identify a novel convergence behavior: the expected number of iterations diminishes with increasing word length. A strong relationship is confirmed using statistical tests to verify a logarithmic relationship, which is also verified using regression modeling and goodness-of-fit tests. In addition to the initial problem statement, this formulation introduces graph pruning as a viable paradigm for feedback-driven inference with interpretability and its association with symbolic reasoning and interactive intelligent systems.

The Traveling Salesperson Problem (TSP) is one of the best-known problems in computer science and arises in many engineering applications, such as smart vehicles and intelligent transportation systems. In the "Euclidean" case, each node is defined by its coordinates in the plane and distances are computed using the Euclidean metric. In the Constraint Programming (CP) literature, the Euclidean TSP is typically addressed by computing the full distance matrix and treating it as a general case; however this approach ignores the geometric information carried by the points' coordinates. In this work, we propose new filtering algorithms, implemented in Constraint Logic Programming (CLP), that exploit such geometric information to achieve stronger constraint propagation than existing approaches. Moreover, we show how this methodology can be extended to other Euclidean variants of the TSP, including the Euclidean Generalized Traveling Salesperson Problem (EGTSP), which is relevant in practical routing and logistics applications. Experimental results demonstrate the computational advantages of the proposed approach.

Chain-of-thought (CoT) prompting can fail severely on constraint-dense logical reasoning tasks, where unverified errors accumulate silently across steps. We introduce SymStep: an LLM makes one atomic claim at a time (DEDUCE: Alice, pet, Cat), then a lightweight constraint propagator checks the claim for consistency with prior accepted deductions, rejects contradictions, and cascades implied facts automatically. SymStep+G additionally provides MRV guidance after each accepted step, directing the LLM toward the most constrained unresolved variable. On a 35-puzzle retained subset of ZebraLogicBench, a benchmark of 1,000 Einstein-style logic puzzles, Direct and CoT both achieve 0%, while SymStep+G reaches 97%. On AR-LSAT analytical reasoning problems, SymStep achieves 100% vs. CoT's 87%. On LGP-14, SymStep+G achieves 100% vs. 0% for CoT and Logic-LM, the strongest prior symbolic+LLM baseline we compare against. Ablation studies reveal that MRV guidance is a key mechanism for reducing directionless cycling, while consistency checking provides a safety net against explicit contradictions. Across six benchmarks spanning five task domains, SymStep variants match or exceed every baseline on constraint-dense and arithmetic tasks. Experiments on AQUA-RAT algebra confirm the advantage is constraint-density-specific.

Difference constraints of the form $x - y \leq d$ are well studied, with efficient algorithms for satisfaction and implication, because of their connection to shortest paths. Finite domain propagation algorithms, however, typically do not make use of these algorithms, and treat each difference constraint as a separate propagator. Propagation does guarantee completeness of solving, but can be needlessly slow. In this paper we describe how to build a (bounds consistent) global propagator for difference constraints that treats them all simultaneously. SAT modulo theory solvers have included theory solvers for difference constraints for some time. While a theory solver for difference constraints gives the basis of a global difference constraint propagator, we show how the requirements on the propagator are quite different. Crucially, we show how to explain propagations by a global difference constraint propagator, in order to use it within a lazy clause generation solver. We give experiments showing that treating difference constraints globally can substantially improve on the standard propagation approach.

One of the main strengths of Constraint Programming is the ability to reduce the search space via propagation. However, propagation is a double-edged sword, with more pruning power coming at the price of larger computation time. For each problem constraint, the best propagator depends on the specific instance and may change at search time. In the literature, Machine Learning (ML) techniques and activity-based heuristics have been applied respectively for choosing (statically) the propagators for a batch of problems and to adapt (dynamically) the propagation strength. We propose to merge those efforts by using an oracle function, obtained via ML, to decide whether to run complex propagators for a target constraint. A combination of design choices makes the approach flexible and easy to embed in state-of-the-art solvers. In this paper, we focus on investigating the feasibility of building an oracle for the Energetic Reasoning propagator. Our experiments show that high prediction accuracy can be obtained, provide suggestions for classification features, and highlight important issues to address when building such an oracle.

Mixed-integer programming (MIP) research is both mathematically sophisticated and engineering-intensive: testing an algorithmic hypothesis within a branch-and-cut solver requires substantial implementation, debugging, tuning, and large-scale benchmarking. We propose an agentic MIP research framework that shortens this feedback loop by embedding LLM agents into a solver-aware harness for generating, verifying, and evaluating plugins for the open-source solver SCIP. Propagation methods play a central role in accelerating MIP solving by exploiting global constraints. We instantiate our framework on the semantic lifting of MIP formulations into global constraints and the automatic construction of propagation-only SCIP constraint handlers. On the MIPLIB 2017 benchmark set, the framework successfully recovers global constraint structures from constraint programming and generates executable constraint detectors and propagation-only constraint handlers. Furthermore, the framework naturally extends to in-context learning within a sandboxed environment, enabling agents not only to tune and debug generated constraint handlers on real instances, but also to explore global constraint patterns in MIP problems and discover novel propagation strategies not yet implemented in SCIP. This framework allows us to systematically distinguish meaningful algorithmic improvements from low-value or overly costly candidates: the novel propagation methods successfully solved five additional instances within the explored benchmark. Overall, this framework demonstrates that LLM agents can autonomously navigate the complex MIP research loop, paving the way for a more automated solver development process.

There are two prevalent model-based paradigms for combinatorial problems: 1) state-based representations, such as heuristic search, dynamic programming (DP), and decision diagrams, and 2) constraint and domain-based representations, such as constraint programming (CP), (mixed-)integer programming, and Boolean satisfiability. In this paper, we bridge the gap between the DP and CP paradigms by integrating constraint propagation into DP, enabling a DP solver to prune states and transitions using constraint propagation. To this end, we implement constraint propagation using a general-purpose CP solver in the Domain-Independent Dynamic Programming framework and evaluate using heuristic search on three combinatorial optimisation problems: Single Machine Scheduling with Time Windows, the Resource Constrained Project Scheduling Problem (RCPSP), and the Travelling Salesperson Problem with Time Windows (TSPTW). Our evaluation shows that constraint propagation significantly reduces the number of state expansions, causing our approach to solve more instances than a DP solver for Single Machine Scheduling and RCPSP, and showing similar improvements for tightly constrained TSPTW instances. The runtime performance indicates that the benefits of propagation outweigh the overhead for constrained instances, but that further work into reducing propagation overhead could improve performance further. Our work is a key step in understanding the value of constraint propagation in DP solvers, providing a model-based approach to integrating DP and CP.

Cumulative constraints are central in scheduling with constraint programming, yet propagation is typically performed per constraint, missing multi-resource interactions and causing severe slowdowns on some benchmarks. I present a preprocessing method for inferring additional cumulative constraints that capture such interactions without search-time probing. This approach interprets cumulative constraints as linear inequalities over occupancy vectors and generates valid inequalities by (i) discovering covers, the sets of tasks that cannot run in parallel, (ii) strengthening the cover inequalities for the discovered sets with lifting, and (iii) injecting the resulting constraints back into the scheduling problem instance. Experiments on standard RCPSP and RCPSP/max test suites show that these inferred constraints improve search performance and tighten objective bounds on favorable instances, while incurring little degradation on unfavorable ones. Additionally, these experiments discover 25 new lower bounds and five new best solutions; eight of the lower bounds are obtained directly from the inferred constraints.

Modeling scheduling problems with conditional time intervals and cumulative functions has become a common approach when using modern commercial constraint programming solvers. This paradigm enables the modeling of a wide range of scheduling problems, including those involving producers and consumers. However, it is unavailable in existing open-source solvers and practical implementation details remain undocumented. In this work, we present an implementation of this modeling approach using a single, generic global constraint called the Generalized Cumulative. We also introduce a novel timetabling filtering algorithm specifically designed to handle tasks defined on conditional time-intervals. Experimental results demonstrate that this approach, combined with the new filtering algorithm, performs competitively with existing solvers enabling the modeling of producer and consumer scheduling problems and effectively scales to large-scale problems.

TCSPs (Temporal Constraint Satisfaction Problems) [Dechter et al. 1991] get rid of unary constraints by binarizing them after having added an "origin of the world" variable. In this work, we look at the constraints between the "origin of the world" variable and the other variables, as the (binarized) domains of these other variables. With this in mind, we define a notion of arc-consistency for TCSPs, which we will refer to as binarized-domains Arc-Consistency, or bdArc-Consistency for short. We provide an algorithm achieving bdArc-Consistency for a TCSP, which we will refer to as bdAC-3, for it is an adaptation of Mackworth's [1977] well-known arc-consistency algorithm AC-3. We show that if an STP is bdArc-Consistent, and connected, i.e., its "origin of the world" variable is disconnected from none of the other variables, its binarized domains are minimal. We provide two polynomial backtrack-free procedures: one for the task of getting a solution from a connected bdArc-Consistent STP; the other for the task of getting, from a bdArc-Consistent STP, either that it is inconsistent or, in case of consistency, a connected bdArc-Consistent STP refinement. We then show how to use our results both in a general TCSP solver and in a TCSP-based job shop scheduler. The work also provides an experimental comparison on STPs of bdAC-3 with an existing arc-consistency algorithm, ACSTP, restricted to STPs [Kong et al. 2018]; an experimental comparison of three TCSP-based job shop schedulers, two of which use weak versions of bdAC-3 as the filtering procedure during the search, the third [Schwalb and Dechter 1997] a weak version of path-consistency; and the swi-prolog source codes used by these comparisons. Last but not least, we provide an incremental version of bdAC-3.