### Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics [chapter]

Stefan Klootwijk, Bodo Manthey, Sander K. Visser
2018 Lecture Notes in Computer Science
A graph G = (V, E) satisfies the α, β-cut-property if the fraction of edges present in each cut of the graph lies between α and β. The Erdős-Rényi random graph G(n, p) satisfies this property w.h.p. for α = (1 − ε)p and β = (1 + ε)p whenever p is sufficiently large and ε is a suitably chosen constant. We study the behavior of random shortest path metrics applied to graphs G that satisfy the α, β-cut-property. These random metrics are defined as follows: Let w(e) be independently drawn random
more » ... e weights for all e ∈ E, and define d (u, v) to be the shortest path distance between u and v in G with respect to the weights w. Using the ideas of Bringmann et al. (Algorithmica, 2015 ) , who studied random shortest path metrics on the complete graph, i.e., the graph that satisfies the 1, 1-cut-property, we derive some properties of the metric and obtain a clustering of the vertices. Using this, we conduct a probabilistic analysis of some simple heuristics on these random shortest path metrics.