On the Complexity and Approximation of the Min-Sum and Min-Max Disjoint Paths Problems

Abstract

Given a graph G = (V, E) and k source-sink pairs {(s 1, t 1), ..., (s k , t k )} with each s i , t i ∈ V, the Min-Sum Disjoint Paths problem asks k disjoint paths to connect all the source-sink pairs with minimized total length, while the Min-Max Disjoint Paths problem asks also k disjoint paths to connect all source-sink pairs but with minimized length of the longest path. In this paper we show that the weighted Min-Sum Disjoint Paths problem is FP NP-complete in general graph, and the uniform Min-Sum Disjoint Paths and uniform Min-Max Disjoint Paths problems can not be approximated within Ω(m 1 − ε) for any constant ε> 0 even in planar graph if P ≠ NP, where m is the number of edges in G. Then we give at the first time a simple bicriteria approximation algorithm for the uniform Min-Max Edge-Disjoint Paths and the weighted Min-Sum Edge-Disjoint Paths problems, with guaranteed performance ratio O(logk / loglogk) and O(1) respectively. Our algorithm is based on randomized rounding.