2-Approximation for Prize-Collecting Steiner Forest

TL;DR

The paper studies the prize-collecting Steiner forest problem (PCSF), a generalization of Steiner forest with penalties for unmet demands. It develops a combinatorial, coloring-based framework that yields a deterministic polynomial-time $2$-approximation, improving the prior $2.54$-approximation and circumventing the known integrality-gap barrier of the natural LP. The core method blends static and dynamic coloring, supported by a SetPairGraph and max-flow computations to bridge colorings with pair demands, first achieving a $3$-approximation (PCSF3) and then an iterative scheme (IPCSF) that attains the $2$-approximation, with prospects for a tighter bound approaching $2-rac{1}{n}$. This advances the practical understanding of PCSF and aligns its approximation quality with that of the Steiner forest problem, highlighting a robust, purely combinatorial approach to prize-collecting network design.

Abstract

Approximation algorithms for the prize-collecting Steiner forest problem (PCSF) have been a subject of research for over three decades, starting with the seminal works of Agrawal, Klein, and Ravi and Goemans and Williamson on Steiner forest and prize-collecting problems. In this paper, we propose and analyze a natural deterministic algorithm for PCSF that achieves a $2$-approximate solution in polynomial time. This represents a significant improvement compared to the previously best known algorithm with a $2.54$-approximation factor developed by Hajiaghayi and Jain in 2006. Furthermore, K{ö}nemann, Olver, Pashkovich, Ravi, Swamy, and Vygen have established an integrality gap of at least $9/4$ for the natural LP relaxation for PCSF. However, we surpass this gap through the utilization of a combinatorial algorithm and a novel analysis technique. Since $2$ is the best known approximation guarantee for Steiner forest problem, which is a special case of PCSF, our result matches this factor and closes the gap between the Steiner forest problem and its generalized version, PCSF.

Figures (5)

• Figure 1: Illustration of the static coloring and the coloring used for Steiner forest problem. In the graph, $S_2$ is inactive and does not color its cutting edges, while $S_1$ colors edges in red and $S_3$ colors in blue. It is worth noting that edges within a connected component will not be further colored and will not be added to $F$.
• Figure 2: SetPairGraph
• Figure 3: By choosing a small positive value $\epsilon < \min(f, \pi_{i'j'}-f")$, we can remove one tight pair. The red variables represent the amounts of flow on each edge, while the black variables represent their capacity.
• Figure 4: A comparison between a single-cut set (left) and a multi-cut set (right).
• Figure 5: The figure shows the graph of $F^*$ with pairs $(i, j)$ and $(i', j')$, and a single-edge set colored with pair $(i,j)$ in dynamic coloring. Tightness of $(i, j)$ implies tightness of $(i', j')$, and removing edge $e$ does not disconnect pairs in $\mathcal{CC}$.

Theorems & Definitions (73)

• Theorem 1
• Definition 2: MaxFlow
• Definition 3: Static coloring duration
• Corollary 4
• Lemma 5
• proof
• Lemma 6
• proof
• Definition 7: Dynamic Coloring Assignment Duration
• Definition 8: Dynamic Coloring Duration