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

In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G = (V,E), edge costs {ce φ 0}e2E, terminal pairs {(si, ti)}ki =1, and penalties {i}ki =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 i. The Steiner forest problem can be viewed as the special case where i = 1 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