TY - GEN

T1 - Lowest degree κ-Spanner

T2 - 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014

AU - Chlamtáč, Eden

AU - Dinitz, Michael

N1 - Publisher Copyright:
© Eden Chlamtáč and Michael Dinitz.

PY - 2014/9/1

Y1 - 2014/9/1

N2 - A κ-spanner is a subgraph in which distances are approximately preserved, up to some given stretch factor κ. We focus on the following problem: Given a graph and a value κ, can we find a κ-spanner that minimizes the maximum degree? While reasonably strong bounds are known for some spanner problems, they almost all involve minimizing the total number of edges. Switching the objective to the degree introduces significant new challenges, and currently the only known approximation bound is an Õ (δ3-2√2)-approximation for the special case when κ = 2 [Chlamtáč, Dinitz, Krauthgamer FOCS 2012] (where δ is the maximum degree in the input graph). In this paper we give the first non-trivial algorithm and polynomial-factor hardness of approximation for the case of general k. Specifically, we give an LP-based Õ(δ(1-1/k)2 )-approximation and prove that it is hard to approximate the optimum to within δ ω (1/k) when the graph is undirected, and to within δ ω(1) when it is directed.

AB - A κ-spanner is a subgraph in which distances are approximately preserved, up to some given stretch factor κ. We focus on the following problem: Given a graph and a value κ, can we find a κ-spanner that minimizes the maximum degree? While reasonably strong bounds are known for some spanner problems, they almost all involve minimizing the total number of edges. Switching the objective to the degree introduces significant new challenges, and currently the only known approximation bound is an Õ (δ3-2√2)-approximation for the special case when κ = 2 [Chlamtáč, Dinitz, Krauthgamer FOCS 2012] (where δ is the maximum degree in the input graph). In this paper we give the first non-trivial algorithm and polynomial-factor hardness of approximation for the case of general k. Specifically, we give an LP-based Õ(δ(1-1/k)2 )-approximation and prove that it is hard to approximate the optimum to within δ ω (1/k) when the graph is undirected, and to within δ ω(1) when it is directed.

KW - Approximation algorithms

KW - Graph spanners

KW - Hardness of approximation

UR - http://www.scopus.com/inward/record.url?scp=84920162912&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2014.80

DO - 10.4230/LIPIcs.APPROX-RANDOM.2014.80

M3 - Conference contribution

AN - SCOPUS:84920162912

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 80

EP - 95

BT - Leibniz International Proceedings in Informatics, LIPIcs

A2 - Rolim, Jose D. P.

A2 - Moore, Cristopher

A2 - Devanur, Nikhil R.

A2 - Jansen, Klaus

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 4 September 2014 through 6 September 2014

ER -