Min-Sum 2-Paths Problems

Trevor Fenner, Oded Lachish, Alexandru Popa

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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 (si,ti), 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 si to ti. 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 si and ti 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.

Original languageEnglish
Pages (from-to)94-110
Number of pages17
JournalTheory of Computing Systems
Volume58
Issue number1
DOIs
Publication statusPublished - Jan 1 2016

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Keywords

  • Algorithms
  • Graph
  • Min-sum
  • Orientation
  • Paths

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

Fenner, T., Lachish, O., & Popa, A. (2016). Min-Sum 2-Paths Problems. Theory of Computing Systems, 58(1), 94-110. https://doi.org/10.1007/s00224-014-9569-1