A Game Theoretic Approach for Efficient Graph Coloring

• Authors: Panagiota N. Panagopoulou;Paul G. Spirakis
• Venue: ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
• Year: 2008

We give an efficient local search algorithm that computes a goodvertex coloring of a graph G. In order to betterillustrate this local search method, we view local moves as selfishmoves in a suitably defined game. In particular, given a graphG = (V,E) of n vertices andm edges, we define the graph coloring gameΓ(G) as a strategic game where the set ofplayers is the set of vertices and the players share the sameaction set, which is a set of n colors. The payoff that avertex v receives, given the actions chosen by allvertices, equals the total number of vertices that have chosen thesame color as v, unless a neighbor of v has alsochosen the same color, in which case the payoff of v is 0.We show:The game Γ(G) has always pure Nashequilibria. Each pure equilibrium is a proper coloring ofG. Furthermore, there exists a pure equilibrium thatcorresponds to an optimum coloring.We give a polynomial time algorithm $\mathcal{A}$ whichcomputes a pure Nash equilibrium ofΓ(G).The total number, k, of colors used in anypure 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 numberof G and Δ 2 is the maximumdegree that a vertex can have subject to the condition that it isadjacent to at least one vertex of equal or greater degree.(Δ 2 is no more than the maximum degree” of G.)Thus, in fact, we propose here a new, efficient coloring methodthat achieves a number of colors satisfying (together) the knowngeneral upper bounds on the chromatic number X.Our method is also an alternative general way of proving,constructively, all these bounds.Finally, we show how to strengthen our method (staying inpolynomial time) so that it avoids "bad" pure Nash equilibria (i.e.those admitting a number of colors k far away fromX). In particular, we show that our enhancedmethod colors optimally dense random q-partitegraphs (of fixed q) with high probability.