$\ell_p$-norm Multiway Cut

TL;DR

This paper introduces the $\,\ell_p$-norm multiway cut problem, unifying classic graph partitioning problems, and provides approximation algorithms, hardness results, and integrality gap analysis for it.

Contribution

It generalizes multiway cut problems using the $\,\ell_p$-norm, offers an $O(\log^2 n)$-approximation, and establishes hardness and integrality gaps.

Findings

NP-hardness for constant terminals and planar graphs

$O(\log^2 n)$-approximation algorithm for all $p\ge 1$

Integrality gap of $\Omega(k^{1-1/p})$ and inapproximability results

Abstract

We introduce and study $\ell_p$-norm-multiway-cut: the input here is an undirected graph with non-negative edge weights along with $k$ terminals and the goal is to find a partition of the vertex set into $k$ parts each containing exactly one terminal so as to minimize the $\ell_p$-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when $p=1$) and min-max multiway cut (when $p=\infty$), both of which are well-studied classic problems in the graph partitioning literature. We show that $\ell_p$-norm-multiway-cut is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an $O(\log^2 n)$-approximation for all $p\ge 1$. We also show an integrality gap of $\Omega(k^{1-1/p})$ for a natural convex program and an $O(k^{1-1/p-\epsilon})$-inapproximability for any constant $\epsilon>0$ assuming the small set expansion hypothesis.