Optimal Approximations for the Requirement Cut Problem on Sparse Graph Classes

TL;DR

This paper improves approximation algorithms for the Requirement Cut problem on certain sparse graph classes by leveraging structural parameters, achieving single-logarithmic ratios where previous methods had double-logarithmic bounds.

Contribution

It introduces new approximation algorithms for Requirement Cut on graphs with limited Steiner trees and bounded depth series-parallel graphs, extending classes with single-logarithmic ratios.

Findings

LP-rounding yields an $O(\log n)$-approximation when Steiner trees are polynomial.

A probabilistic embedding provides an $O(depth.\log(g))$-approximation for bounded-depth series-parallel graphs.

Results extend single-logarithmic approximation ratios to new graph classes.

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.