Approximability Distance in the Space of H-Colourability Problems

TL;DR

This paper introduces a metric on the space of graphs to extend approximation results for the MAX H-Colourable Subgraph problem, connecting it to known algorithms for MAX CUT and MAX k-CUT, and compares algorithm performances.

Contribution

It develops a metric framework on graph space to generalize approximation results for MAX H-COL, linking it to classical algorithms and conjectures.

Findings

Approximation algorithms for MAX CUT and MAX k-CUT can be applied to MAX H-COL.

Near-optimality results are shown under the Unique Games Conjecture.

Frieze & Jerrum's algorithm generally outperforms Hastad's for MAX 2-CSP.

Abstract

A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. We study the approximability properties of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL). The instances of this problem are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H=K_k this problem is equivalent to MAX k-CUT. To this end, we introduce a metric structure on the space of graphs which allows us to extend previously known approximability results to larger classes of graphs. Specifically, the approximation algorithms for MAX CUT by Goemans and Williamson and MAX k-CUT by Frieze and Jerrum can be used to yield non-trivial approximation results for MAX H-COL. For a variety of graphs, we show near-optimality results under the Unique Games Conjecture. We also use our method for comparing the performance of Frieze & Jerrum's algorithm with Hastad's approximation algorithm for general MAX 2-CSP. This comparison is, in most cases, favourable to Frieze & Jerrum.