Successive shortest paths in complete graphs with random edge weights

Stefanie Gerke,Gregory B Sorkin,Balázs F Mezei

doi:10.1002/rsa.20962

Abstract

Consider a complete graph Kn with edge weights drawn independently from a uniform distribution U(0, 1). The weight of the shortest (minimum‐weight) path P1 between two given vertices is known to be , asymptotically. Define a second‐shortest path P2 to be the shortest path edge‐disjoint from P1, and consider more generally the shortest path Pk edge‐disjoint from all earlier paths. We show that the cost Xk of Pk converges in probability to uniformly for all k ≤ n − 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapest k edge‐disjoint paths, that is, a minimum‐cost k‐flow. We also obtain the expectation of Xk conditioned on the existence of Pk.

Highlights

It is a standard problem to find the shortest s–t path in a graph, i.e., the cheapest pathP1 between specified vertices s and t, and its cost X1, where the cost of a path is the sum of the costs of its edges
The novel element here is the “robustness” aspect of finding cheap structures even after the cheapest has been removed, and in this we were motivated by a recent study by Janson and Sorkin [JS19] of the same question for successive minimum spanning trees (MSTs), again for Kn with uniform or exponential random edge weights
Upper bound for small k we prove the upper bound of Theorem 1.1 for all k = o(√n); larger values are treated

Summary

Introduction

It is a standard problem to find the shortest s–t path in a graph, i.e., the cheapest path. The novel element here is the “robustness” aspect of finding cheap structures even after the cheapest has been removed, and in this we were motivated by a recent study by Janson and Sorkin [JS19] of the same question for successive minimum spanning trees (MSTs), again for Kn with uniform or exponential random edge weights. In many contexts (including for the length X1 of a shortest path) the result for any distribution with positive density at 0 follows immediately from that for the uniform distribution U (0, 1), but that is not the case for the successive paths considered here. It is not clear for what other edge-weight distributions (even those with density 1 at 0) (1.4) will hold

Open problems

Edge order statistics

Lower bound

Exponential model

CB 4 Cε

Expectation