Equitable Coloring of Graphs

Furmańczyk, Hanna

doi:10.1090/conm/352/03

Cited by 36 publications

(43 citation statements)

References 7 publications

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“…Due to load balancing considerations, it is desirable to perform equal or nearly-equal numbers of tasks in each time slot, and this balancing is exactly what equitable colorings achieve. Furmańczyk [5] mentions a specific application of this type of scheduling problem, namely, assigning university courses to time slots in a way that avoids scheduling incompatible courses at the same time and spreads the courses evenly among the available time slots.…”

Section: Introductionmentioning

confidence: 99%

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Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

Furmańczyk

Kubale

2016

Ars Math. Contemp.

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A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ = (G). In this paper the problem of determinig χ = for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses χ = (G) or χ = (G) + 1 colors. Our algorithm is best possible, unless P=NP. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring.

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Section: Introductionmentioning

confidence: 99%

“…It was considered for some particular graph classes and also for several graph products: cartesian, weak or strong tensor products [13,5] as well as for coronas [6,10]. Graph products are interesting and useful in many situations.…”

Section: Introductionmentioning

confidence: 99%

Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

Furmańczyk

Kubale

2016

Ars Math. Contemp.

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“…Specifically, Furmanczyk and Kubale proposed two constructive heuristics (Naive and Subgraph) [11]. Bahiense et al presented two effective branchand-cut algorithms [1,2].…”

Section: Introductionmentioning

confidence: 99%

Backtracking based iterated tabu search for equitable coloring

Lai

Hao

Glover

2015

Engineering Applications of Artificial Intelligence

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An equitable k-coloring of an undirected graph G = (V, E) is a partition of its vertices into k disjoint independent sets, such that the cardinalities of any two independent sets differ by at most one. As a variant of the graph coloring problem (GCP), the equitable coloring problem (ECP) concerns finding a minimum k for which an equitable k-coloring exists. In this work, we propose a backtracking based iterated tabu search (BITS) algorithm for solving the ECP approximately. BITS uses a backtracking scheme to define different k-ECP instances, an iterated tabu search approach to solve each particular k-ECP instance for a fixed k, and a binary search approach to find a suitable initial value of k. We assess the algorithm's performance on a set of commonly used benchmarks. Computational results show that BITS is very competitive in terms of solution quality and computing efficiency compared to the state-of-the-art algorithm in the literature. Specifically, BITS obtains new upper bounds for 21 benchmark instances, while matching the previous best upper bound for the remaining instances. Finally, to better understand the proposed algorithm, we study how its key ingredients impact its performance.

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“…Other applications of the ECP concern load balancing problems in multiprocessor machines [3] and results in probability theory [4]. An introduction to ECP and some basic results are provided in [5].…”

Section: Introductionmentioning

confidence: 99%

“…Two constructive heuristics called Naive and SubGraph are given in [5] to generate greedily an equitable coloring of a graph. There also exist heuristic algorithms for constructing colorings that are "nearly" equitable [8,9], making emphasis on achieving a small difference between the sizes of the biggest class and the smallest one, although the equity constraint still might be violated.…”

Section: Introductionmentioning

confidence: 99%

A Tabu Search Heuristic for the Equitable Coloring Problem

Díaz¹,

Nasini²,

Severín³

2014

Lecture Notes in Computer Science

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Abstract. The Equitable Coloring Problem is a variant of the GraphColoring Problem where the sizes of two arbitrary color classes differ in at most one unit. This additional condition, called equity constraints, arises naturally in several applications. Due to the hardness of the problem, current exact algorithms can not solve large-sized instances. Such instances must be addressed only via heuristic methods. In this paper we present a tabu search heuristic for the Equitable Coloring Problem. This algorithm is an adaptation of the dynamic TabuCol version of Galinier and Hao. In order to satisfy equity constraints, new local search criteria are given. Computational experiments are carried out in order to find the best combination of parameters involved in the dynamic tenure of the heuristic. Finally, we show the good performance of our heuristic over known benchmark instances.

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Equitable Coloring of Graphs

Cited by 36 publications

References 7 publications

Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

Backtracking based iterated tabu search for equitable coloring

A Tabu Search Heuristic for the Equitable Coloring Problem

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