Approximate Max-Flow Min-Multicut Theorem for Graphs of Bounded Treewidth

TL;DR

This paper establishes an approximate max-multiflow min-multicut theorem for graphs with bounded treewidth, providing a tight bound and a polynomial-time approximation algorithm with logarithmic factor for the multicut problem.

Contribution

It proves a tight approximate max-multiflow min-multicut theorem for bounded treewidth graphs and introduces a constructive polynomial-time approximation algorithm.

Findings

The multicommodity flow and multicut capacities are within an O(log r) factor for treewidth-r graphs.

The result is tight up to constant factors, matching known lower bounds.

Provides a polynomial-time O(log r)-approximation algorithm for the multicut problem.

Abstract

We prove an approximate max-multiflow min-multicut theorem for bounded treewidth graphs. In particular, we show the following: Given a treewidth-$r$ graph, there exists a (fractional) multicommodity flow of value $f$, and a multicut of capacity $c$ such that $ f \leq c \leq \mathcal{O}(\ln (r+1)) \cdot f$. It is well known that the multiflow-multicut gap on an $r$-vertex (constant degree) expander graph can be $\Omega(\ln r)$, and hence our result is tight up to constant factors. Our proof is constructive, and we also obtain a polynomial time $\mathcal{O}(\ln (r+1))$-approximation algorithm for the minimum multicut problem on treewidth-$r$ graphs. Our algorithm proceeds by rounding the optimal fractional solution to the natural linear programming relaxation of the multicut problem. We introduce novel modifications to the well-known region growing algorithm to facilitate the rounding while guaranteeing at most a logarithmic factor loss in the treewidth.