# On Approximate Graph Colouring and MAX-*k*-CUT Algorithms Based on the θ-Function

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## Abstract

The problem of colouring a *k*-colourable graph is well-known to be NP-complete, for *k* ≥ 3. The MAX-*k*-CUT approach to approximate *k*-colouring is to assign *k* colours to all of the vertices in polynomial time such that the fraction of `defect edges' (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and Jerrum (1997), using a semidefinite programming (SDP) relaxation which is related to the Lovász θ-function. In a related work, Karger et al. (1998) devised approximation algorithms for colouring *k*-colourable graphs exactly in polynomial time with as few colours as possible. They also used an SDP relaxation related to the θ-function.

In this paper we further explore semidefinite programming relaxations where graph colouring is viewed as a satisfiability problem, as considered in De Klerk et al. (2000). We first show that the approximation to the chromatic number suggested in De Klerk et al. (2000) is bounded from above by the Lovász θ-function. The underlying semidefinite programming relaxation in De Klerk et al. (2000) involves a lifting of the approximation space, which in turn suggests a provably good MAX-*k*-CUT algorithm. We show that of our algorithm is closely related to that of Frieze and Jerrum; thus we can sharpen their approximation guarantees for MAX-*k*-CUT for small fixed values of *k*. For example, if *k* = 3 we can improve their bound from 0.832718 to 0.836008, and for *k* = 4 from 0.850301 to 0.857487. We also give a new asymptotic analysis of the Frieze-Jerrum rounding scheme, that provides a unifying proof of the main results of both Frieze and Jerrum (1997) and Karger et al. (1998) for *k* ≫ 0.

*k*-CUT

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