  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 on complete graphs to analyze
heuristics.
  The goal of this paper is to generalize these findings to non-complete
graphs, especially Erd\H{o}s-R\'enyi 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.
