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-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.
| Original language | English |
|---|---|
| Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
| Publisher | Springer Verlag |
| Pages | 1-11 |
| Number of pages | 11 |
| Volume | 8447 LNCS |
| ISBN (Print) | 9783319080000 |
| DOIs | |
| Publication status | Published - 2014 |
| Externally published | Yes |
| Event | 11th International Workshop on Approximation and Online Algorithms, WAOA 2013 - Sophia Antipolis, France Duration: Sep 5 2013 → Sep 6 2013 |
Publication series
| Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
|---|---|
| Volume | 8447 LNCS |
| ISSN (Print) | 03029743 |
| ISSN (Electronic) | 16113349 |
Other
| Other | 11th International Workshop on Approximation and Online Algorithms, WAOA 2013 |
|---|---|
| Country | France |
| City | Sophia Antipolis |
| Period | 9/5/13 → 9/6/13 |
Fingerprint
ASJC Scopus subject areas
- Computer Science(all)
- Theoretical Computer Science
Cite this
Min-sum 2-paths problems. / Fenner, Trevor; Lachish, Oded; Popa, Alexandru.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 8447 LNCS Springer Verlag, 2014. p. 1-11 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8447 LNCS).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
TY - GEN
T1 - Min-sum 2-paths problems
AU - Fenner, Trevor
AU - Lachish, Oded
AU - Popa, Alexandru
PY - 2014
Y1 - 2014
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.
UR - http://www.scopus.com/inward/record.url?scp=84903635435&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84903635435&partnerID=8YFLogxK
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
VL - 8447 LNCS
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 1
EP - 11
BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PB - Springer Verlag
ER -