Approximation Algorithms for Norm Multiway Cut

TL;DR

This paper develops improved approximation algorithms for the generalized Norm Multiway Cut problem, extending classic graph partitioning with norm-based objectives and analyzing oracle-based and hardness scenarios.

Contribution

It introduces new approximation algorithms for Norm Multiway Cut with different oracle assumptions and improves bounds for the $ extit{ ext{l}_p}$-norm Multiway problem.

Findings

Improved $O( ext{log}^{1/2} n ext{log}^{1/2+1/p} k)$ approximation for $ extit{ ext{l}_p}$-norm Multiway.

New approximation algorithms for Norm Multiway Cut with varying oracle access.

Hardness results assuming the Hypergraph Dense-vs-Random Conjecture.

Abstract

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph $G$ into $k$ parts so as to separate $k$ given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced $\ell_p$-norm Multiway, a generalization of the problem, in which the goal is to minimize the $\ell_p$ norm of the edge boundaries of $k$ parts. We provide an $O(\log^{1/2} n\log^{1/2+1/p} k)$ approximation algorithm for this problem, improving upon the approximation guarantee of $O(\log^{3/2} n \log^{1/2} k)$ due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an $O(\log^{1/2} n \log^{7/2} k)$ approximation algorithm with a weaker oracle and an $O(\log^{1/2} n \log^{5/2} k)$ approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no $n^{1/4-\varepsilon}$ approximation algorithm for every $\varepsilon > 0$ assuming the Hypergraph Dense-vs-Random Conjecture.