TY - JOUR
T1 - Min-Sum 2-Paths Problems
AU - Fenner, Trevor
AU - Lachish, Oded
AU - Popa, Alexandru
N1 - Publisher Copyright:
© 2014, Springer Science+Business Media New York.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.
PY - 2016/1/1
Y1 - 2016/1/1
N2 - 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.
AB - 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.
KW - Algorithms
KW - Graph
KW - Min-sum
KW - Orientation
KW - Paths
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U2 - 10.1007/s00224-014-9569-1
DO - 10.1007/s00224-014-9569-1
M3 - Article
AN - SCOPUS:84952944101
VL - 58
SP - 94
EP - 110
JO - Theory of Computing Systems
JF - Theory of Computing Systems
SN - 1432-4350
IS - 1
ER -