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 {(s(1), t(1)), ... , (s(k), t(k))} with each s(i), t(i) epsilon 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 source-sink 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 Paths problem cannot be approximated within Omega(m(1-epsilon)) for any constant epsilon > 0 even in planar graphs, assuming P not equal 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.