Min-Sum 2-Paths Problems

An orientation of an undirected graph G is a directed graph obtained by replacing each edge {u,v} of G by exactly one of the arcs (u,v) or (v,u). In the min-sumk-paths orientation problem, the input is an undirected graph G and ordered pairs (s_i,t_i), where i∈{1,2,…,k}. The goal is to find an orientation of G that minimizes the sum over all i∈{1,2,…,k} of the distance from s_i to t_i. In the min-sumkedge-disjoint paths problem, the input is the same, however the goal is to find for every i∈{1,2,…,k} a path between s_i and t_i so that these paths are edge-disjoint and the sum of their lengths is minimum. Note that, for every fixed k≥2, the question of NP-hardness for the min-sum k-paths orientation problem and for the min-sum k edge-disjoint paths problem has been open for more than two decades. We study the complexity of these problems when k=2. We exhibit a PTAS for the min-sum 2-paths orientation problem. A by-product of this PTAS is a reduction from the min-sum 2-paths orientation problem to the min-sum 2 edge-disjoint paths problem. The implications of this reduction are: (i) an NP-hardness proof for the min-sum 2-paths orientation problem yields an NP-hardness proof for the min-sum 2 edge-disjoint paths problem, and (ii) any approximation algorithm for the min-sum 2 edge-disjoint paths problem can be used to construct an approximation algorithm for the min-sum 2-paths orientation problem with the same approximation guarantee and only an additive polynomial increase in the running time.