Optimal Approximations for the Requirement Cut Problem on Sparse Graph Classes

TL;DR

The paper advances single-log approximations for the Requirement Cut problem by identifying two structural graph parameters that govern tractability: the number of minimal Steiner trees $\sigma(G,\mathcal{S})$ and the depth of series-parallel graphs. It proves that if $\sigma(G,\mathcal{S})$ is polynomial in $n$, a simple LP-rounding yields an $O(\log n)$-approximation; if the input is a series-parallel graph of constant depth, a refined analysis of existing tree-embedding methods delivers an $O(\text{depth} \cdot \log(g))$-approximation. The core technique combines LP relaxations with randomized rounding and a carefully constructed equivalence between Steiner-tree cuts and RC feasibility, alongside an embedding-based argument that leverages graph depth. Together, these results extend single-log approximability beyond trees to broader graph classes and illuminate how Steiner-tree counts and SP-depth influence approximation performance for a unified framework of cut problems.

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.

Figures (4)

• Figure 1: Hierarchy of graph cut problems captured by the Requirement Cut framework putting our results in perspective. Boxes contain parameters, and best known approximation results for general graphs (and for more particular cases when specified). Arrows denote special cases of the problems (i.e., the source is a particular case of the target).
• Figure 2: Overview of graph classes and instances for which the Requirement Cut problem admits a single-logarithmic approximation in this paper.
• Figure 3: This graph has $\mathcal{O}(\log(n))$ cycles of length 3. each spanning tree chooses 1 edge per cycle to remove, hence yielding a number of spanning trees in $\mathcal{O}(3^{\log(n)})$ which is polynomial $\mathcal{O}(n^{\log(3)})$.
• Figure 4: Instance where terminal groups reside in minimally connected subgraphs. Here, $\tau(H_i) = poly(|H_i|), i\in\{1,2,3\}$ would suffice to have a $\log(n)$ approximation for the instance.

Theorems & Definitions (7)

• definition thmcounterdefinition: Cut
• definition thmcounterdefinition: Requirement Cut Instance
• definition thmcounterdefinition: Steiner Tree
• definition thmcounterdefinition: Minimal Steiner Tree
• definition thmcounterdefinition: Series-Parallel Graphs
• definition thmcounterdefinition: Composition Trace
• definition thmcounterdefinition: Depth of a Series-Parallel Graph