Complexity of Coloring Random Graphs

Abstract

It is known that the problem of deciding k -colorability of a graph exhibits an easy-hard-easy pattern,—that is, the average-case complexity for backtrack-type algorithms, as a function of k , has a peak. This complexity peak is either at k = χ − 1 or k = χ, where χ is the chromatic number of the graph. However, the behavior around the complexity peak is poorly understood. In this article, we use list coloring to model coloring with a fractional number of colors between χ − 1 and χ. We present a comprehensive computational study on the complexity of backtrack-type graph coloring algorithms in this critical range. According to our findings, an easy-hard-easy pattern can be observed on a finer scale between χ − 1 and χ as well. The highest complexity found this way can be higher than for any integer value of k . It turns out that the complexity follows an alternating three-dimensional pattern; understanding this pattern is very important for benchmarking purposes. Our results also answer the previously open question whether coloring with χ − 1 or with χ colors is harder: this depends on the location of the maximal fractional complexity.