License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.17
URN: urn:nbn:de:0030-drops-75665
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7566/
Könemann, Jochen ;
Olver, Neil ;
Pashkovich, Kanstantsin ;
Ravi, R. ;
Swamy, Chaitanya ;
Vygen, Jens
On the Integrality Gap of the Prize-Collecting Steiner Forest LP
Abstract
In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G=(V,E), nonnegative edge costs {c_e} for e in E, terminal pairs {(s_i,t_i)} for i=1,...,k, and penalties {pi_i} for i=1,...,k for each terminal pair; the goal is to find a forest F to minimize c(F) + sum{ pi_i: (s_i,t_i) is not connected in F }. The Steiner forest problem can be viewed as the special case where pi_i are infinite for all i. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF (PCSF-LP) is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least 9/4 even if the input instance is planar. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than 4. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP-approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most 1/3 and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than 3 using a direct iterative rounding method.
BibTeX - Entry
@InProceedings{knemann_et_al:LIPIcs:2017:7566,
author = {Jochen K{\"o}nemann and Neil Olver and Kanstantsin Pashkovich and R. Ravi and Chaitanya Swamy and Jens Vygen},
title = {{On the Integrality Gap of the Prize-Collecting Steiner Forest LP}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {17:1--17:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7566},
URN = {urn:nbn:de:0030-drops-75665},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.17},
annote = {Keywords: Integrality gap, Steiner tree, Steiner forest, prize-collecting, Lagrangianmultiplier- preserving}
}
|
Keywords: |
|
Integrality gap, Steiner tree, Steiner forest, prize-collecting, Lagrangianmultiplier- preserving |
|
Collection: |
|
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017) |
|
Issue Date: |
|
2017 |
|
Date of publication: |
|
11.08.2017 |
