perform competitive analysis

This work investigates how to distinguish “easy” from “hard” input distributions in the stochastic caching problem to overcome the limitations imposed by worst-case analysis. To this end, we introduce subset entropy—a concept from information theory—as a novel parameter that enables the first fine-grained quantification of input distribution complexity. Building upon this measure, we develop a unified analytical framework applicable to both online and stochastic optimization settings. Within this framework, we establish competitive ratio upper bounds for classical algorithms such as LRU that explicitly depend on subset entropy. Our results demonstrate that under low-entropy distributions, these algorithms achieve substantially better performance than their classical worst-case guarantees, thereby providing new theoretical justification for the empirical effectiveness of LRU under realistic, structured inputs.

This paper studies the prediction-augmented online caching problem, where each request is accompanied by a prediction of the next occurrence time of the corresponding page. Methodologically, we first improve the upper bound on the competitive ratio of the BlindOracle algorithm. Second, we establish, for the first time, a lower bound of Ω(√h) on the competitive ratio of any randomized online caching algorithm, where h denotes the prediction error bound—thereby revealing the inherent hardness of the problem. Third, we propose a hybrid strategy combining BlindOracle and Marker, achieving an O(1)-competitive ratio under bounded prediction error, which is optimal up to constant factors. Collectively, our results significantly tighten both the upper and lower bounds on the competitive ratio for prediction-augmented caching, yielding the strongest known theoretical guarantees to date. This work provides a foundational benchmark for learning-augmented online algorithms.

This work addresses the long-standing bottleneck in prophet inequality research—manual construction of worst-case instances for tight competitive ratio proofs—by proposing the first unified framework that directly models tightness analysis as a computationally tractable optimization problem. Methodologically, it formalizes worst-case instance search as a convex optimization and linear programming problem subject to extremal probability distribution constraints, integrating tools from random-order theory and extremal probability analysis, thereby replacing the traditional decoupled paradigm of “algorithm analysis + counterexample construction.” Contributions include: (i) the first automated computation of tight competitive ratios; (ii) a unified derivation of tight bounds for multiple prophet inequality variants, yielding several new results; and (iii) rigorous verification of the optimality of several classical bounds. The framework significantly enhances the systematicity, scalability, and reliability of tightness proofs in online stochastic optimization.

This paper studies the online unit-density knapsack problem: heterogeneous-sized items arrive sequentially and must be irrevocably accepted or rejected upon arrival (no splitting), aiming to maximize total accepted size. The problem models real-time inventory allocation under unpredictable order streams in wholesale supply chains. We propose the first systematic framework for stochastic competitive ratio analysis and design an optimal randomized threshold policy: for a single knapsack, it achieves a tight 0.432-competitive ratio—matching the theoretical optimum; extended to multiple knapsacks, it attains a 0.214-competitive ratio—the best-known and tight bound to date. Our results establish the first nontrivial stochastic lower bound for the untruncated AdWords problem and significantly advance the theoretical frontier of online knapsack-type problems.

This work addresses the lack of a unified methodology in competitive analysis of online algorithms by proposing a posterior matching principle grounded in a minimax perspective. Leveraging Yao’s principle, it reframes worst-case competitive analysis as a Bayesian online design problem with an arbitrary correlated prior, selecting at each step the feasible action closest to the posterior expected optimum. The framework integrates Bayesian inference, posterior process tracking, martingale analysis, and randomized rounding, establishing a connection between information theory and competitive ratios via vector martingale inequalities. It achieves or surpasses the best-known competitive ratios for fractional problems—including set cover, load balancing, matching, and generalized resource allocation—and provides, for the first time, theoretical guarantees for randomized algorithms on integer problems such as weighted paging, star-metric metrical task systems, and ski rental.

This work addresses the online problem of chasing sets of size at most $k$ in metric spaces—equivalently, layered graph traversal with width $k$. By generalizing the classical doubling strategy, it presents the first deterministic online algorithm achieving an $O(2^k)$ competitive ratio against adaptive adversaries. The paper establishes the tight deterministic competitive ratio for this problem as $\Theta(2^k)$, demonstrates that the generalized Work Function Algorithm is suboptimal in this setting, and introduces a novel recursive lower-bound construction $D_k$. Notably, matching upper and lower bounds are provided for the case $k=3$, leading to improved bounds for related problems such as distributed asynchronous tree exploration and the $k$-taxi problem.