Improved Combinatorial Approximation Algorithms for MAX CUT in Sparse Graphs

TL;DR

This paper introduces improved combinatorial approximation algorithms for the Max-Cut problem in sparse graphs, achieving better ratios and solving an open problem in subcubic graphs using a novel graph decomposition method.

Contribution

It presents a new vertex decomposition called tree-bipartite decomposition and develops linear-time algorithms with improved approximation ratios for Max-Cut in sparse and subcubic graphs.

Findings

Linear-time $(rac{1}{2}+rac{n-1}{2m})$-approximation algorithm for Max-Cut in sparse graphs.

Linear-time $rac{5}{6}$-approximation algorithm for Max-Cut in subcubic graphs.

Solves an open problem regarding approximation ratios in subcubic graphs.

Abstract

The Max-Cut problem is a fundamental NP-hard problem, which is attracting attention in the field of quantum computation these days. Regarding the approximation algorithm of the Max-Cut problem, algorithms based on semidefinite programming have achieved much better approximation ratios than combinatorial algorithms. Therefore, filling the gap is an interesting topic as combinatorial algorithms also have some merits. In sparse graphs, there is a linear-time combinatorial algorithm with the approximation ratio $\frac{1}{2}+\frac{n-1}{4m}$ [Ngoc and Tuza, Comb. Probab. Comput. 1993], which is known as the Edwards-Erd\H{o}s bound. In subcubic graphs, the combinatorial algorithm by Bazgan and Tuza [Discrete Math. 2008] has the best approximation ratio $\frac{5}{6}$ that runs in $O(n^2)$ time. Based on the approach by Bazgan and Tuza, we introduce a new vertex decomposition of graphs, which we call tree-bipartite decomposition. With the decomposition, we present a linear-time $(\frac{1}{2}+\frac{n-1}{2m})$-approximation algorithm for the Max-Cut problem. As a derivative, we also present a linear-time $\frac{5}{6}$-approximation algorithm in subcubic graphs, which solves an open problem in their paper.