A flow based pruning scheme for enumerative equitable coloring algorithms

TL;DR

This paper introduces a flow-based pruning scheme for enumerative equitable coloring algorithms, significantly reducing search tree size and improving computational efficiency through flow and arithmetic pruning rules.

Contribution

It presents a novel network flow model to derive pruning rules for equitable coloring, enhancing the efficiency of enumerative algorithms.

Findings

Reduced search tree size in algorithms

Faster computation of equitable chromatic number

Improved stability and solved instances within time limits

Abstract

An equitable graph coloring is a proper vertex coloring of a graph G where the sizes of the color classes differ by at most one. The equitable chromatic number is the smallest number k such that G admits such equitable k-coloring. We focus on enumerative algorithms for the computation of the equitable coloring number and propose a general scheme to derive pruning rules for them: We show how the extendability of a partial coloring into an equitable coloring can be modeled via network flows. Thus, we obtain pruning rules which can be checked via flow algorithms. Computational experiments show that the search tree of enumerative algorithms can be significantly reduced in size by these rules and, in most instances, such naive approach even yields a faster algorithm. Moreover, the stability, i.e., the number of solved instances within a given time limit, is greatly improved. Since the execution of flow algorithms at each node of a search tree is time consuming, we derive arithmetic pruning rules (generalized Hall-conditions) from the network model. Adding these rules to an enumerative algorithm yields an even larger runtime improvement.