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 language | English |
|---|---|
| Pages (from-to) | 94-110 |
| Number of pages | 17 |
| Journal | Theory of Computing Systems |
| Volume | 58 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Jan 1 2016 |
| Externally published | Yes |
Fingerprint
Keywords
- Algorithms
- Graph
- Min-sum
- Orientation
- Paths
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Theory and Mathematics
Cite this
Min-Sum 2-Paths Problems. / Fenner, Trevor; Lachish, Oded; Popa, Alexandru.
In: Theory of Computing Systems, Vol. 58, No. 1, 01.01.2016, p. 94-110.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Min-Sum 2-Paths Problems
AU - Fenner, Trevor
AU - Lachish, Oded
AU - Popa, Alexandru
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
UR - http://www.scopus.com/inward/record.url?scp=84952944101&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84952944101&partnerID=8YFLogxK
U2 - 10.1007/s00224-014-9569-1
DO - 10.1007/s00224-014-9569-1
M3 - Article
VL - 58
SP - 94
EP - 110
JO - Theory of Computing Systems
JF - Theory of Computing Systems
SN - 1432-4350
IS - 1
ER -