TY - GEN

T1 - Min-sum 2-paths problems

AU - Fenner, Trevor

AU - Lachish, Oded

AU - Popa, Alexandru

PY - 2014/1/1

Y1 - 2014/1/1

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-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 every 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 NP-hardness for the min-sum k-paths orientation problem and the min-sum k edge-disjoint paths problem have 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-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 every 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 NP-hardness for the min-sum k-paths orientation problem and the min-sum k edge-disjoint paths problem have 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.

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U2 - 10.1007/978-3-319-08001-7_1

DO - 10.1007/978-3-319-08001-7_1

M3 - Conference contribution

AN - SCOPUS:84903635435

SN - 9783319080000

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 11

BT - Approximation and Online Algorithms - 11th International Workshop, WAOA 2013, Revised Selected Papers

PB - Springer Verlag

T2 - 11th International Workshop on Approximation and Online Algorithms, WAOA 2013

Y2 - 5 September 2013 through 6 September 2013

ER -