Prize-Collecting Steiner Tree: A 1.79 Approximation

TL;DR

The paper addresses Prize-Collecting Steiner Tree (PCST), where each vertex bears a penalty and one must decide whether to connect it or pay the penalty. It introduces IPCST, an iterative, polynomial-time algorithm that combines a Goemans–Williamson baseline on penalized instances with a Steiner Tree subroutine and recursive penalty adjustments, ultimately selecting the best of three candidate trees. By carefully selecting the Steiner Tree factor $p$ and penalty scale $\beta$, the authors establish a $1.7994$-approximation with $\alpha=1.7994$, $\beta=1.252$, and $p=\ln(4)+\epsilon$, improving upon the long-standing barrier below $2$ for PCST. The approach leverages rooted PCST equivalence, reduces to Steiner Tree subproblems on live vertices, and runs in polynomial time, offering significant implications for practical network design and related prize-collecting problems.

Abstract

Prize-Collecting Steiner Tree (PCST) is a generalization of the Steiner Tree problem, a fundamental problem in computer science. In the classic Steiner Tree problem, we aim to connect a set of vertices known as terminals using the minimum-weight tree in a given weighted graph. In this generalized version, each vertex has a penalty, and there is flexibility to decide whether to connect each vertex or pay its associated penalty, making the problem more realistic and practical. Both the Steiner Tree problem and its Prize-Collecting version had long-standing $2$-approximation algorithms, matching the integrality gap of the natural LP formulations for both. This barrier for both problems has been surpassed, with algorithms achieving approximation factors below $2$. While research on the Steiner Tree problem has led to a series of reductions in the approximation ratio below $2$, culminating in a $\ln(4)+ε$ approximation by Byrka, Grandoni, Rothvoß, and Sanità, the Prize-Collecting version has not seen improvements in the past 15 years since the work of Archer, Bateni, Hajiaghayi, and Karloff, which reduced the approximation factor for this problem from $2$ to $1.9672$. Interestingly, even the Prize-Collecting TSP approximation, which was first improved below $2$ in the same paper, has seen several advancements since then. In this paper, we reduce the approximation factor for the PCST problem substantially to 1.7994 via a novel iterative approach.

Figures (4)

• Figure 1: Illustration of dead sets in the final tree of GW algorithm. The dead sets colored in blue cut multiple edges of $F$, and removing them would disconnect other vertices so they are not removed. On the other hand, the dead sets colored in red can be safely removed without affecting other vertices.
• Figure 2: An illustration of how the components in $C$ and $C'$ can be. The components in $C'$ are shown in red circles, and the components in $C$ are shown in blue ones. Each red component that does not include $\textit{root}$ is also a blue component.
• Figure 3: Illustration of single-edge set vs. multi-edge set in $T_{\text{OPT}}$. The red set is a single-edge set, but the blue one is a multi-edge set.
• Figure 4: A star graph with $n+1$ vertices. We construct a PCST instance on this graph with vertex $r$ as the root, the central vertex $c$ having penalty $0$, and all other vertices with having penalty $2(1+\frac{1}{n-1})$.

Theorems & Definitions (41)

• Definition 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• Definition 7
• Definition 8: Connected and unconnected dead vertices
• Definition 9: Single-edge and multi-edge sets