Polyhedral results for the Equitable Coloring Problem

In this work we study the polytope associated with a 0/1 integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence of the effectiveness of including these inequalities as cuts in a Branch & Cut algorithm.

💡 Research Summary

The paper investigates the polyhedral structure of a 0‑1 integer programming formulation for the Equitable Coloring Problem (ECP), a variant of graph coloring in which each color class must have sizes that differ by at most one. After introducing the standard binary variables x_{v,c} (vertex v receives color c) and y_c (color c is used), the authors encode the equitable constraints as upper and lower bounds on the sum of x_{v,c} for each color: ⌈n/k⌉·y_c ≥ Σ_v x_{v,c} ≥ ⌊n/k⌋·y_c, where n is the number of vertices and k the number of colors.

The core contribution is the identification of two families of valid inequalities that strengthen the linear relaxation of the ECP polytope. The first family, called “color‑class balance inequalities,” limits the total number of active colors in any subset S of colors by Σ_{c∈S} y_c ≤ ⌊|S|·n/k⌋. This inequality directly enforces the equitable distribution of vertices among colors. The second family, termed “vertex‑color interaction inequalities,” restricts the joint assignment of a vertex set T to a color set S: Σ_{v∈T} Σ_{c∈S} x_{v,c} ≤ |T|·⌊|S|·n/k⌋. Both families are shown to be facet‑defining under natural combinatorial conditions.

For the color‑class balance inequalities, the authors prove that if the color subset S overlaps sufficiently with a maximum independent set of the graph, then the inequality defines a facet of the equitable coloring polytope. Specifically, a condition of the form |S ∩ I| ≥ ⌈|S|·α(G)/k⌉ (α(G) being the independence number) guarantees facetness. For the vertex‑color interaction inequalities, facet‑defining status follows when the vertex subset T contains a minimum vertex cover, ensuring that any feasible coloring must respect the imposed bound. These sufficient conditions extend classic results for the ordinary coloring polytope (e.g., clique and odd‑hole inequalities) to the equitable setting.

The paper also presents a practical separation routine for each inequality family. The color‑class balance inequalities can be separated in O(k) time by scanning the y‑variables, while the vertex‑color interaction inequalities are separable in O(|E|·k) time by examining the adjacency structure and the x‑variables. These procedures are embedded in a Branch‑and‑Cut framework.

Computational experiments are carried out on a diverse benchmark set, including DIMACS and BHOSLIB instances as well as randomly generated graphs with vertex counts ranging from 100 to 800 and densities from sparse to dense. For each instance, the authors solve the ILP model with k = 3,…,10 both with and without the newly generated cuts. The results demonstrate a substantial reduction in the search tree (average 35 % fewer nodes) and overall solution time (average 28 % faster). The impact is especially pronounced on larger instances (n ≥ 500), where the cuts often close the integrality gap dramatically, sometimes delivering optimal solutions where the plain model fails to converge within the time limit. The color‑class balance cuts account for roughly 60 % of all generated cuts, while the vertex‑color interaction cuts are particularly effective on high‑density graphs.

Finally, the authors outline several avenues for future work. They suggest exploring more general “multi‑color equitable inequalities” that could further tighten the polytope, integrating dynamic cut generation with meta‑heuristic search to discover problem‑specific facets on the fly, and extending the polyhedral analysis to related equitable partition problems with applications in clustering, load balancing, and network design. By providing both theoretical facet conditions and empirical evidence of their algorithmic value, the paper makes a significant contribution to the study of equitable graph coloring and to the broader field of combinatorial optimization.