On the Slow Convergence to Trivial Solutions of Algorithms for Hard Optimization Problems

2026-08-19

AI Summary

研究了难优化问题算法缓慢收敛于平凡解的问题，通过分析和实验表明，在有限规模下，局部算法表现优于理论预测。

Abstract

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.

Problem

• combinatorial optimization
• algorithmic performance
• convergence
• constraint density
• asymptotic behavior

Innovation

• finite-size behavior
• local algorithms
• asymptotic theory
• optimization performance