A Game Theoretic Approach for Efficient Graph Coloring

Abstract

We give an efficient local search algorithm that computes a good vertex coloring of a graph G. In order to better illustrate this local search method, we view local moves as selfish moves in a suitably defined game. In particular, given a graph G = (V,E) of n vertices and m edges, we define the graph coloring game Γ(G) as a strategic game where the set of players is the set of vertices and the players share the same action set, which is a set of n colors. The payoff that a vertex v receives, given the actions chosen by all vertices, equals the total number of vertices that have chosen the same color as v, unless a neighbor of v has also chosen the same color, in which case the payoff of v is 0. We show:

• The game Γ(G) has always pure Nash equilibria. Each pure equilibrium is a proper coloring of G. Furthermore, there exists a pure equilibrium that corresponds to an optimum coloring.

• We give a polynomial time algorithm $\mathcal{A}$ which computes a pure Nash equilibrium of Γ(G).

• The total number, k, of colors used in any pure Nash equilibrium (and thus achieved by $\mathcal{A}$) is $k\leq\min\{\Delta_2+1, \frac{n+\omega}{2}, \frac{1+\sqrt{1+8m}}{2}, n-\alpha+1\}$, where ω, α are the clique number and the independence number of G and Δ ₂ is the maximum degree that a vertex can have subject to the condition that it is adjacent to at least one vertex of equal or greater degree. (Δ ₂ is no more than the maximum degree Δ of G.)

• Thus, in fact, we propose here a new, efficient coloring method that achieves a number of colors satisfying (together) the known general upper bounds on the chromatic number χ. Our method is also an alternative general way of proving, constructively, all these bounds.

• Finally, we show how to strengthen our method (staying in polynomial time) so that it avoids “bad” pure Nash equilibria (i.e. those admitting a number of colors k far away from χ). In particular, we show that our enhanced method colors optimally dense random q-partite graphs (of fixed q) with high probability.

Partially supported by the EU within the 6th Framework Programme under contract 015964 “Algorithmic Principles for Building Efficient Overlay Computers” (AEOLUS) and the ICT Programme under contract IST-2008-215270 (FRONTS), and by the General Secretariat for Research and Technology of the Greek Ministry of Development within the Programme PENED 2003.