Prize-Collecting Forest with Submodular Penalties: Improved Approximation

TL;DR

This work addresses the prize-collecting forest problem (PCF) on weighted graphs with submodular penalties for unmet connectivity demands, and achieves a deterministic polynomial-time 2-approximation. Building on the 3-approximation framework of Sharma, Swamy, and Williamson (SSW), the authors devise an iterative, penalty-based recursion that defines marginal penalties $\pi_R$ to progressively refine the solution while paying tight-set penalties. They prove, via a careful case analysis that partitions tight sets into $\mathcal{D}_{paid}$ and $\mathcal{D}_{unpaid}$ (further split into $\mathcal{D}_{single}$ and $\mathcal{D}_{multiple}$), and by induction on recursion, that the returned forest cost is at most $2\cdot\text{OPT}$. The approach leverages submodular minimization in the separation oracles and extends Goemans–Williamson style techniques to a highly general model, showing that submodular penalties can be integrated without sacrificing a 2-approximation. This yields a broadly applicable improvement for prize-collecting and generalized forest problems, with implications for Steiner trees/forests and related connectivity settings where penalties are submodular and defined over exponential domains.

Abstract

Constrained forest problems form a class of graph problems where specific connectivity requirements for certain cuts within the graph must be satisfied by selecting the minimum-cost set of edges. The prize-collecting version of these problems introduces flexibility by allowing penalties to be paid to ignore some connectivity requirements. Goemans and Williamson introduced a general technique and developed a 2-approximation algorithm for constrained forest problems. Further, Sharma, Swamy, and Williamson extended this work by developing a 2.54-approximation algorithm for the prize-collecting version of these problems. Motivated by the generality of their framework, which includes problems such as Steiner trees, Steiner forests, and their variants, we pursued further exploration. We present a significant improvement by achieving a 2-approximation algorithm for this general model, matching the approximation factor of the constrained forest problems.

Figures (3)

• Figure 1: Tight sets returned by SSW, denoted by $\mathcal{D}_\text{SSW}$, are partitioned into $\mathcal{D}_{paid}$ and $\mathcal{D}_{unpaid}$, with $\mathcal{D}_{unpaid}$ further divided into $\mathcal{D}_{single}$ and $\mathcal{D}_{multiple}$.
• Figure 2: An illustration of the steps in Lemma \ref{['lm:rforestpen']}. When removing the sole edge $e$ cut by set $\textcolor{Red}{S}$, component $\textcolor{Black}{C}$ is split into components $\textcolor{Blue}{C_1}$ and $\textcolor{Green}{C_2}$. It can be seen that any vertex in $\textcolor{Green}{C_2}$ being in $\textcolor{Red}{S}$ or any vertex in $\textcolor{Blue}{C_1}$ being outside $\textcolor{Red}{S}$ will lead to another edge cut by $\textcolor{Red}{S}$. Therefore, $\textcolor{Red}{S} \cap \textcolor{Black}{C}$ will be $\textcolor{Blue}{C_1}$, as shown here.
• Figure 3: This figure illustrates how adding sets into ${\color{Blue} \mathcal{D}_{base}}$, affects the implicit partition ${\color{Green} M_{base}}$. Initially, ${\color{Blue} \mathcal{D}_{base}}$ and ${\color{Green} M_{base}}$ are both $\{V\}$. $S_i$ represents sets added to ${\color{Blue} \mathcal{D}_{base}}$ at step $i$. As shown here, the number of sets in ${\color{Green} M_{base}}$ increases by at least one in each step, since the partial intersections of the newly added $S_i$ with sets in ${\color{Green} M_{base}}$ create smaller minimal sets. Furthermore, in each step, for any set $S$ in $\textsc{closure}({\color{Green} M_{base}})$, $\pi(S\mid{\color{Blue} \mathcal{D}_{base}}) = 0$. For example, for family $\mathcal{X} = \{\{A\}, \{B\}, \{A, B\}\}$, we have $\pi^{(2)}(\mathcal{X}) = \pi^{(1)}(\mathcal{X} \mid \{S_2\}) = \pi(\mathcal{X} \mid \{S_1, S_2\}) = 0$.

Theorems & Definitions (53)

• Theorem 1
• Definition 2
• Lemma 3
• proof
• Definition 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7