Complexity and Approximation Results for the Min-Sum and Min-Max Disjoint Paths Problems

Given a graph G = (V,E) and k source-sink pairs {(s1, t1), . . . , (sk, tk)} with each si, ti ∈ V , the Min-Sum Disjoint Paths problem asks to find k disjoint paths connecting all the source-sink pairs with minimized total length, while the Min-Max Disjoint Paths problem asks for k disjoint paths connecting all the sourcesink pairs with minimized length of the longest path. We show that the weighted Min-Sum Disjoint Paths problem is FP-complete in general graphs, and the unweighted Min-Sum Disjoint Paths problem and the unweighted Min-Max Disjoint 24 P. Zhang, W. Zhao, D. Zhu Paths problem cannot be approximated within Ω(m1− ) for any constant > 0 even in planar graphs, assuming P 6= NP, where m is the number of edges in G. We give for the first time a simple bicriteria approximation algorithm for the unweighted Min-Max Edge-Disjoint Paths problem and the weighted Min-Sum Edge-Disjoint Paths problem, with guaranteed approximation ratio O(log k/ log log k) and O(1), respectively.