Coloring the Vertices of 9-pt and 27-pt Stencils with Intervals

Abstract

Graph coloring is commonly used to schedule computations on parallel systems. Given a good estimation of the computational requirement for each task, one can refine the model by adding a weight to each vertex. Instead of coloring each vertex with a single color, the problem is to color each vertex with an interval of colors. In this paper, we are interested in studying this problem for particular classes of graphs, namely stencil graphs. Stencil graphs appear naturally in the parallelisation of applications where the location of an object in a space affects the state of neighboring objects. Rectilinear decompositions of a space generate conflict graphs that are 9-pt stencils for 2D problems and 27-pt stencils for 3D problems. We show that the 5-pt stencil and 7-pt stencil relaxations of the problem can be solved in polynomial time. We prove that the decision problem on 27-pt stencil is NP-Complete. We discuss approximation algorithms with a ratio of 2 for the 9-pt stencil case, and 4 for the 27-pt stencil case. We identify two lower bounds for the problem that are used to design heuristics. We evaluate the effectiveness of several different algorithms experimentally on a set of real instances. Furthermore, these algorithms are integrated into a real application to demonstrate the soundness of the approach.