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On the integrality gap of the prize-collecting steiner forest LP

Könemann, Jochen, Olver, Neil, Pashkovich, Kanstantsin, Ravi, R, Swamy, Chaitanya and Vygen, Jens (2017) On the integrality gap of the prize-collecting steiner forest LP. In: Approx 2017 - Random 2017, 2017-08-16 - 2017-08-18, UC Berkley.

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Identification Number: 10.4230/LIPIcs.APPROX-RANDOM.2017.17


In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G = (V, E), edge costs {ce ≥ 0}e∈E, terminal pairs {(si , ti)} k i=1, and penalties {πi} k i=1 for each terminal pair; the goal is to find a forest F to minimize c(F) + P i:(si,ti) not connected in F πi . The Steiner forest problem can be viewed as the special case where πi = ∞ 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. This holds even for planar graphs. 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.

Item Type: Conference or Workshop Item (Paper)
Official URL:
Additional Information: © 2017 The Author(s)
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Jan 2020 10:06
Last Modified: 20 Jul 2021 00:33

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