Optimal Approximations for the Requirement Cut Problem on Sparse Graph Classes

2025-05-27

AI Summary

This paper studies the Requirement Cut problem—a unifying generalization of Multicut, Multiway Cut, $k$-Cut, and Steiner Multicut. Given an edge-weighted graph, terminal groups $(S_1,dots,S_g)$, and demands $(r_1,dots,r_g)$, the goal is to find a minimum-cost edge cut such that each $S_i$ is separated into at least $r_i$ connected components. The previous best approximation ratio was $O(log g cdot log n)$, achieved via tree embeddings. We introduce two novel structural parameters: the number of minimum Steiner trees (polynomially bounded) and the series-parallel depth. Combining LP rounding, probabilistic embedding, Steiner tree enumeration, and structural graph decomposition, we achieve single-logarithmic approximations: $O(log n)$ for graphs with polynomially many minimum Steiner trees, and $O( ext{depth} cdot log g)$ for series-parallel graphs of constant depth. These results significantly broaden the class of graphs admitting $O(log n)$-approximations.

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: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

Abstract

We study the Requirement Cut problem, a generalization of numerous classical graph partitioning problems including Multicut, Multiway Cut, $k$-Cut, and Steiner Multicut among others. Given a graph with edge costs, terminal groups $(S_1, ..., S_g)$ and integer requirements $(r_1,... , r_g)$; the goal is to compute a minimum-cost edge cut that separates each group $S_i$ into at least $r_i$ connected components. Despite many efforts, the best known approximation for Requirement Cut yields a double-logarithmic $O(log(g).log(n))$ approximation ratio as it relies on embedding general graphs into trees and solving the tree instance. In this paper, we explore two largely unstudied structural parameters in order to obtain single-logarithmic approximation ratios: (1) the number of minimal Steiner trees in the instance, which in particular is upper-bounded by the number of spanning trees of the graphs multiplied by $g$, and (2) the depth of series-parallel graphs. Specifically, we show that if the number of minimal Steiner trees is polynomial in $n$, then a simple LP-rounding algorithm yields an $O(log n)$-approximation, and if the graph is series-parallel with a constant depth then a refined analysis of a known probabilistic embedding yields a $O(depth.log(g))$-approximation on series-parallel graphs of bounded depth. Both results extend the known class of graphs that have a single-logarithmic approximation ratio.

Problem

• Improves approximation for Requirement Cut on sparse graphs
• Explores Steiner trees and series-parallel depth parameters
• Achieves single-logarithmic ratios via LP-rounding and embeddings

Innovation

• LP-rounding for polynomial Steiner trees
• Probabilistic embedding for series-parallel graphs
• Single-logarithmic approximation via structural parameters