Probabilistic analysis of optimization problems on generalized random shortest path metrics

Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis falls short of explaining this performance. Because this, “beyond worst-case analysis” algorithms has recently gained lot attention, including probabilistic algorithms. The instances many problems are essentially discrete metric space. Probabilistic such nevertheless mostly been conducted on drawn from Euclidean space, which provides structure that is usually heavily exploited the analysis. However, most not Euclidean. Little work done other, more realistic, distributions. Some initial results have obtained by Bringmann et al. (Algorithmica, 2013), who used random shortest path metrics constructed using complete graphs to analyze heuristics. goal paper generalize these findings non-complete graphs, especially Erdős–Rényi graphs. A drawing independent edge weights each graph and setting distance between every pair vertices length them with respect weights. For instances, we prove greedy heuristic minimum maximum matching problem, nearest neighbor insertion traveling salesman trivial k-median problem all achieve constant expected approximation ratio. Additionally, polynomial upper bound number iterations 2-opt problem.