Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis often falls short of explaining this performance. Because of this, “beyond worst-case analysis” of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms.

The instances of many optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained by Bringmann et al. (Algorithmica, 2013), who have used random shortest path metrics constructed using complete graphs to analyze heuristics.

The goal of this paper is to generalize these findings to non-complete graphs, especially Erdős–Rényi random graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances, we prove that the greedy heuristic for the minimum distance maximum matching problem, the nearest neighbor and insertion heuristics for the traveling salesman problem, and a trivial heuristic for the k-median problem all achieve a constant expected approximation ratio. Additionally, we show a polynomial upper bound for the expected number of iterations of the 2-opt heuristic for the traveling salesman problem.

简单的启发式方法通常在优化问题上表现出卓越的性能。最坏情况的分析通常无法解释这种性能。因此，算法的“超越最坏情况的分析”最近引起了很多关注，包括算法的概率分析。

许多优化问题的实例本质上是离散的度量空间。然而，针对此类度量优化问题的概率分析主要是在从欧几里得空间中提取的实例上进行的，该实例提供了通常在分析中被大量利用的结构。但是，实践中大多数实例不是欧几里得。对于从其他更实际的分布中提取的度量实例，我们所做的工作很少。Bringmann等人已经获得了一些初步结果。（Algorithmica，2013年），他们使用了随机的最短路径度量标准，该度量标准是使用完整图构建的，用于分析启发式算法。

本文的目的是将这些发现概括为不完整的图，尤其是Erdős-Rényi随机图。随机最短路径度量是通过为图形中的每个边缘绘制独立的随机边缘权重并将每对顶点之间的距离设置为相对于所绘制的权重之间的最短路径的长度来构造的。对于这种情况，我们证明了最小距离最大匹配问题的贪婪启发式，旅行商问题的最近邻和插入启发式以及k中位数问题的琐碎启发式均实现了恒定的预期近似率。此外，我们为旅行商问题展示了2-opt启发式算法的预期迭代次数的多项式上限。