发表机构
LAAS–CNRS; Universidad de San Andrés; SISSA; International Center for Theoretical Physics（LAAS–CNRS; 圣安德烈斯大学; 国际高等研究学院; 国际理论物理中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析难解优化问题算法的有限尺寸行为，发现其收敛到理论界的过程极为缓慢，中间情形下局部算法性能优于渐近预测，表明复杂算法设计仍具重要性。
AI 中文摘要
难解组合优化问题多为NP难问题，构成了基础算法挑战。针对随机实例的平均情形分析已成为一种强大框架，用于理解超出最坏情况保证之外的典型算法性能。大量研究已得出负面结论：对于足够难解的实例（通常由底层图连通性/约束密度控制），在问题规模与约束密度均趋向无穷的双重渐近极限下，尚无已知多项式时间算法能显著优于朴素启发式算法。我们通过研究某些优化算法在易、中、难三种情形下的有限尺寸行为，重新审视这一图景。结合对大图渐近的严格分析与对典型问题（最大独立集和最大K-SAT）的数值实验，我们证明算法虽最终会收敛到理论预测的界，但这种收敛可能极为缓慢。在实例已高度受约束的中间情形中，局部算法能获得远优于高约束密度极限下预测性能的解。这种有限情形与渐近行为间的差距具有重要实践意义：即便渐近理论预测必然失败，复杂的算法设计仍至关重要。
英文摘要
Hard combinatorial optimization problems, many of which are NP-hard, present fundamental algorithmic challenges. Average-case analysis on random instances has emerged as a powerful framework for understanding typical algorithmic performance beyond worst-case guarantees. A substantial body of work has established negative results: for sufficiently hard instances (often controlled by the underlying graph connectivity/constraints density), no known polynomial-time algorithm can significantly outperform naive heuristics in the double asymptotic limit where both problem size and constraints density tend to infinity. We revisit this picture by studying the finite-size behavior of some optimization algorithms across easy, intermediate, and hard regimes. Through rigorous analysis of large-graph asymptotics combined with numerical experiments on canonical problems (maximum independent set and maximum $K$-SAT), we demonstrate that while algorithms do eventually converge to theoretically predicted bounds, this convergence can be remarkably slow. In the intermediate regime where instances are already highly constrained, local algorithms achieve solutions substantially better than their predicted performance in the high-constraint-density limit. This gap between finite-regime and asymptotic behavior has important practical implications: sophisticated algorithmic design remains crucial even when asymptotic theory predicts inevitable failure.
