Competitive Analysis

In the cow-path problem, a cow must find a goal lying at an unknown distance on one of $w$ paths connected only at the origin, and performance is measured by competitive ratio. Kao, Reif and Tate designed an efficient randomized algorithm in which the cow visits the paths in a fixed cyclic order. They proved the algorithm is optimal for $w=2$, and subsequently Kao, Ma, Sipser and Yin proved its optimality for all $w$, with a claim that no algorithm does better than the best cyclic one. This note provides a detailed proof of that claim.

Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.

This paper studies learning-augmented and randomized online aggregation with delays on a line metric. We consider advice given as online suggested service lengths, and evaluate the algorithms in terms of robustness and consistency. For each $λ\in (0,1]$, we first propose a deterministic learning-augmented \textsc{Balance} algorithm that is $(4/λ+1/λ^2)$-robust and $(4+λ)$-consistent. We also propose a randomized algorithm for the problem in the classical adversarial model, which is $(e+1)$-competitive against an oblivious adversary, improving over the deterministic $5$-competitive \textsc{Balance} benchmark~\cite{bienkowski2013chain}. Notably, this competitive ratio is even lower than the lower bound of $4$ for deterministic online algorithms. Moreover, we establish a lower bound of $e$ on the competitive ratio of randomized online algorithms, improving the previous lower bound of $e/(e-1)$. Besides, we combine the two ideas and obtain a randomized learning-augmented algorithm that is $(e/λ+1/λ^2)$-robust and $(e+λ)$-consistent. Finally, we conduct numerical experiments to complement our theoretical analysis and evaluate the empirical performance of our algorithms.

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.

Learning-augmented paging has been extensively studied in recent years. A key advantage over naive ML-based approaches is \emph{bounded robustness}, which guarantees worst-case performance even when predictions are inaccurate, making these algorithms valuable for real-world systems. Prior work achieves robustness bounds of $2H_k + O(1)$ in the randomized setting, leaving a gap to the optimal competitive ratio $H_k$. In this paper, we study how to close this gap. We begin by reviewing online optimality and proving a new property of the latest $H_k$-competitive algorithm, which facilitates our analysis in the learning-augmented setting. Then, we review existing learning-augmented paging algorithms and introduce a unifying primitive, the \emph{relative prediction budget}, which captures the essence of establishing robustness and reveals that prior algorithms either overuse or underutilize predictions. Guided by the above analysis, we develop a new framework that achieves the best-possible robustness up to an additive constant for learning-augmented paging: $H_k + O(1)$. Experiments further demonstrate strong practical performance.

Network routers that enforce Quality-of-Service (QoS) guarantees must decide, at every clock cycle, which expiring packet of information to transmit, even when the value of the packet is unknown until it is processed. We frame this problem as the Online Packet Scheduling with Deadlines (OPSD) problem under Partial Feedback: packets arrive at every clock cycle, with different deadlines, but the weights are only observed after execution. Under a stochastic assumption on the unknown weights, we explore different variants of the OPSD problem with bandit feedback. We establish a connection between our setting and the sleeping bandits problem, and set our learning goal to $α$-regret minimization. We provide algorithms with provable $α$-regret guarantees under different spans of slackness, distinguishing systems allowing for randomization and systems that do not. In every scenario, our algorithms achieve an $α$-regret upper bound of $\widetilde{\mathcal{O}}\left(\sqrt{KT}\right)$, matching the lower bound for the standard bandit setting. In the practically relevant case of $2$-bounded deadline instances, where the deadline is set at most one clock cycle away from the arrival, our deterministic algorithm achieves the provably tightest possible competitive ratio. Remarkably, when the number of distinct packet types $K\ge 2$ is finite, it is possible to break the well-established $Φ= \frac{1+\sqrt{5}}{2}$ competitive ratio barrier and attain a tighter competitive ratio $θ_K$ ranging in $[\sqrt{2}, Φ)$.

We study how an e-commerce firm should make real-time fulfillment decisions in a two-layer distribution network when multi-item customer orders arrive sequentially and future demand is unknown. The central managerial tension is whether to use scarce front distribution center (FDC) inventory to save current fulfillment cost or preserve that inventory for future orders that may be more valuable to serve locally. We formulate an adversarial online model with multiple FDCs, one regional distribution center (RDC), multi-unit multi-item orders, and item-specific and time-varying variable costs. Our theoretical objective is to characterize when simple, interpretable, and implementable fulfillment rules can perform nearly as well as an optimal clairvoyant planner. We develop a family of Gated Priority-based Greedy policies, derive competitive-ratio guarantees under both time-varying and time-invariant cost structures, and establish matching or near-matching lower bounds for any online algorithm. Numerical experiments show that the proposed policies perform strongly relative to generalized myopic and forecast-based benchmarks. The analysis yields managerial guidance on when local inventory should be protected, when splitting orders is worth the fixed-cost burden, and how the relative magnitudes of fixed and variable costs determine the value of more sophisticated optimization.

Competition retrospectives are useful when they explain what a leaderboard measured, how hidden evaluation changed conclusions, and which design patterns were rewarded. We revisit the CODS 2025 \assetopslive{} challenge, a privacy-aware Codabench competition on industrial multi-agent orchestration built on \assetops{}. We combine final rank sheets, a 300-submission server log, 149-team registrations, best-submission exports, the organizer winners report, the companion \assetopslive{} system paper, and verified planning-track source trees. Five results stand out. First, the public planning leaderboard saturates at 72.73%, and richer prompts do not improve that peak. Second, hidden evaluation changes the story: public and private scores correlate moderately in planning ($r{=}0.69$) but negatively in execution ($r{=}{-}0.13$), with several 45.45% public execution systems reaching 63.64% on the hidden set. Third, the \tmatch{} term is numerically almost inert in the official composite -- combined on a 0--1 scale with 0--100 percentage scores, it contributes at most 0.05 points per track, and rescaling would swap the top two teams. Fourth, the competition is operationally account-based but substantively team-based: 149 registered teams reduce to 24 with non-zero public scores and 11 fully ranked, while 52.3% of deduplicated registrations list multiple usernames. Fifth, successful execution methods mostly improve guardrails -- response selection, contamination cleanup, fallback, and context control -- rather than novel agent architectures. These findings identify which behaviors the evaluation rewarded, and motivate scale-aware composites, skill-level diagnostics, and versioned artifact release.

We present a learning-augmented online algorithm for the preemptive FIFO buffer management problem, where packets arrive online to a finite-capacity buffer, must be transmitted in FIFO order, and the algorithm may preemptively discard buffered packets to accommodate future arrivals. Our algorithm simultaneously achieves 1-consistency, η-smoothness, and asymptotic \sqrt{3}-robustness, where ηdenotes the prediction error. Specifically, it attains an optimal competitive ratio of 1 under perfect predictions, degrades smoothly as the prediction error increases, and maintains an asymptotic competitive ratio of \sqrt{3} under arbitrarily inaccurate predictions, matching the best-known worst-case guarantee for the classical online problem, established by Englert and Westermann in 2009 [Algorithmica 53(4): 523-548]. A key technical contribution of our work is the introduction of an \emph{output-based prediction error metric}. Because capacity constraints dictate that only a strictly bounded subset of arriving packets is ultimately transmitted, our metric assesses prediction quality over the resulting optimal schedules rather than the raw input sequences, avoiding artificial error penalties. To guarantee robustness, our algorithm dynamically monitors predictions and executes a \emph{buffer-clearing strategy} upon transitioning to a worst-case fallback mechanism. We prove that the competitive loss incurred by this clearing operation is bounded by an additive capacity constant that vanishes asymptotically. Finally, we show that our algorithm provides a generalized framework for learning-augmented buffer management: substituting the fallback module with any β-competitive online algorithm immediately yields asymptotic β-robustness.

We study cooperative shortest path planning for an unmanned ground vehicle (UGV) assisted by an unmanned aerial vehicle (UAV) scout in environments with unknown road blockages that are only discovered when a robot reaches the damaged point. This formulation generalizes the original Canadian Traveller Problem (CTP), which assumes a single ground vehicle and that the traversability status of all incident edges is revealed upon arrival at a vertex. We first analyze the case where the start and the goal are connected by $k$ disjoint paths, and prove that the worst-case competitive ratio $ρ$ for a single UGV is $2k-1$. With UAV assistance, and under the simplifying assumption of negligible initial transit and deadheading UAV costs, the ratio $ρ$ improves to $2(k-1)\frac{v_G}{v_G + v_A} + 1$, where $v_G$ and $v_A$ denote the UGV and UAV speed, respectively. To address general graphs and non-negligible UAV initial transit and deadheading costs, we present an optimal candidate-path partitioning algorithm that assigns path prefix inspection to the UGV and path suffix inspection to the UAV, and prove the optimality of the UAV inspection strategy on general graphs. We evaluate our algorithm by performing experiments on road networks from the world's 50 most populous cities with randomized blockage locations, and show that the algorithm reduces UGV travel time, with larger improvements as the UAV speed increases.

Cost-per-click (CPC) in paid search is an auction-generated outcome shaped by a competitive landscape that is only partially observable from any single advertiser's history. From 1.66 billion Google Ads log records for a concentrated car-rental market (2021-2023), we construct a weekly panel of 1,811 keyword series over 127 weeks (218,924 keyword-week observations) and build competition-aware proxies from keyword text, CPC trajectories, and geographic market structure. The design combines (i) semantic neighborhoods and a semantic keyword graph from pretrained transformer-based keyword representations, (ii) behavioral neighborhoods from Dynamic Time Warping (DTW) alignment of CPC trajectories, and (iii) geographic-intent covariates capturing localized demand and marketplace heterogeneity. We evaluate these signals both as exogenous covariates and as relational priors in spatiotemporal graph forecasters, benchmarking them against statistical, neural, and time-series foundation-model baselines. The results reveal a clear horizon crossover. At one week, graph-based models achieve the lowest error, reducing sMAPE by 15.1% relative to the strongest classical/ML baseline; at the six- and twelve-week horizons, covariate-augmented foundation models dominate, reducing sMAPE by 22.5% and 27.6%, respectively. The gains concentrate in the high-CPC, high-volatility keywords where forecasting errors are most costly. A falsification battery supports the competition interpretation at the planning horizon: the semantic competition graph outperforms a confounder-matched non-competitive graph by 4.05 sMAPE points, and matched-neighbour and time-shuffled controls show the six-week gains are competition-specific rather than generic smoothing. Together, the findings establish a horizon-dependent competition-aware forecasting design for auction-driven advertising markets under partial observability.