Title: 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 $G=(V,E)$, edge costs $\{c_e\geq 0\}_{e\in E}$, terminal pairs
$\{(s_i,t_i)\}_{i=1}^k$, and penalties $\{\pi_i\}_{i=1}^k$ for each terminal
pair; the goal is to find a forest $F$ to minimize $c(F)+\sum_{i:
(s_i,t_i)\text{ not connected in }F}\pi_i$. The Steiner forest problem can be
viewed as the special case where $\pi_i=\infty$ for all $i$. It was widely
believed that the integrality gap of the natural (and well-studied)
linear-programming (LP) relaxation for PCSF 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.