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.

## Abstract

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: | http://cui.unige.ch/tcs/random-approx/2017/ |

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 |

URI: | http://eprints.lse.ac.uk/id/eprint/103111 |

### Actions (login required)

View Item |