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

GivenagraphG = (V,E) andk source-sinkpairs{(s1, t1), . . . , (sk, tk)} with each si, ti ∈ 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-complete in general graph, and the uniform Min-Sum Disjoint Paths and uniform Min-Max Disjoint Paths problems can not be approximated within Ω(m1− ) for any constant > 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(log k/ log log k) and O(1) respectively. Our algorithm is based on randomized rounding.