More on greedy construction heuristics for the MAX-CUT problem

TL;DR

This paper introduces a novel classification of greedy heuristics for the MAX-CUT problem using a relation tree framework, linking graph algorithms to quantum computing concepts and analyzing their performance on different graph types.

Contribution

It classifies MAX-CUT greedy heuristics into Prim and Kruskal classes via relation trees, connecting to quantum stabilizer formalism and analyzing their empirical performance.

Findings

Prim-class algorithms excel on dense graphs.

Kruskal-class algorithms perform better on sparse graphs.

The ADAPT-Clifford algorithm is a reformulation of SG3.

Abstract

A cut of a graph can be represented in many different ways. Here we propose to represent a cut through a ``relation tree'', which is a spanning tree with signed edges. We show that this picture helps to classify the main greedy heuristics for the maximum cut problem, in analogy with the minimum spanning tree problem. Namely, all versions of the Sahni-Gonzalez~(SG) algorithms could be classified as the Prim class, while various Edge-Contraction~(EC) algorithms are of the Kruskal class. We further elucidate the relation of this framework to the stabilizer formalism in quantum computing, and point out that the recently proposed \textit{ADAPT-Clifford} algorithm is a reformulation of a refined version of the SG algorithm, SG3. Numerical performance of the typical algorithms from the two classes are studied with various kinds of graphs. It turns out that, the Prim-class algorithms perform better for general dense graphs, and the Kruskal-class algorithms performs better when the graphs are sparse enough.