Min-Sum 2-Paths Problems

• Published in: Theory of computing systems Vol. 58; no. 1; pp. 94 - 110
• Main Authors: Fenner, Trevor, Lachish, Oded, Popa, Alexandru
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2016; Springer Nature B.V
• Subjects: Algorithms; Approximation; Byproducts; Computer Science; Graph algorithms; Graph theory; Graphs; Mathematical analysis; Orientation; Polynomials; Reduction; Studies; Theory of Computation; Min-sum; Graph; Algorithms; Paths; Orientation
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-014-9569-1

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-sum k -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-sum k edge-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 N P -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.