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.

Complexity Analysis and Stochastic Convergence of Some Well-known Evolutionary Operators for Solving Graph Coloring Problem

“…GCP is very challenging for genetic algorithms because of its vast solution space. Hence, designing and choosing the right genetic operators in population-based methods are important for the following reasons [53][54][55][56]:…”