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arXiv · 2211.06267

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

Tobias Friedrich·Davis Issac·Nikhil Kumar·Nadym Mallek·Ziena Zeif
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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.

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Tobias Friedrich, Davis Issac, Nikhil Kumar, Nadym Mallek, Ziena Zeif. 2022-11-11. Approximate Max-Flow Min-Multicut Theorem for Graphs of Bounded Treewidth. https://arxiv.org/abs/2211.06267

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