On a reduction of the interval coloring problem to a series of bandwidth coloring problems

Abstract

Given a graph G=(V,E) with strictly positive integer weights ? i on the vertices i?V, an interval coloring of G is a function I that assigns an interval I(i) of ? i consecutive integers (called colors) to each vertex i?V so that I(i)?I(j)=? for all edges {i,j}?E. The interval coloring problem is to determine an interval coloring that uses as few colors as possible. Assuming that a strictly positive integer weight ? ij is associated with each edge {i,j}?E, a bandwidth coloring of G is a function c that assigns an integer (called a color) to each vertex i?V so that |c(i)?c(j)|?? ij for all edges {i,j}?E. The bandwidth coloring problem is to determine a bandwidth coloring with minimum difference between the largest and the smallest colors used. We prove that an optimal solution of the interval coloring problem can be obtained by solving a series of bandwidth coloring problems. Computational experiments demonstrate that such a reduction can help to solve larger instances or to obtain better upper bounds on the optimal solution value of the interval coloring problem.