Prize-Collecting Forest with Submodular Penalties: Improved Approximation

2025-04-21

AI Summary

This paper studies the Prize-Collecting Forest problem with submodular penalties: given a graph with edge costs and a submodular penalty function over vertex subsets, the goal is to find a forest minimizing the sum of edge costs and the penalty incurred by uncovered vertices, subject to partial cut-connectivity constraints. We present the first 2-approximation algorithm for this general model—improving upon the previous best ratio of 2.54 and matching the optimal approximation bound for the basic Constrained Forest problem. Our method extends the Goemans–Williamson two-level LP relaxation framework; we design a novel rounding scheme leveraging the diminishing-return property of submodular functions and introduce a potential-function analysis to simultaneously ensure feasibility and cost control. The resulting algorithm strictly dominates prior approaches and yields tight theoretical guarantees for important special cases—including the Steiner Forest problem—thereby establishing a new paradigm for penalty-based network design.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Social mining and social search on the Web

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.

Problem

• Improving approximation for prize-collecting forest problems
• Matching 2-approximation for constrained forest problems
• Extending framework to Steiner trees and forests

Innovation

• 2-approximation algorithm for constrained forests
• Improved prize-collecting forest with penalties
• General model matching previous approximation factor