An introduction to chromatic sums

Kubicka, Ewa; Schwenk, Allen J.

doi:10.1145/75427.75430

Cited by 104 publications

(79 citation statements)

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“…1) implements the general procedure given in section 2.1. EXSCOL starts by identifying a first largest possible independent set I M (line 4) whose size |I M | is used later to build a pool M of independent sets of that size (lines [5][6][7][8][9][10][11][12][13][14][15]. The search for a new independent set of size |I M | stops when the number of independent sets contained in M reaches a desired threshold (M max ) or when no new independent set of that size is found after p max consecutive tries.…”

Section: The Exscol Algorithmmentioning

confidence: 99%

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An effective heuristic algorithm for sum coloring of graphs

Hao

2012

Computers & Operations Research

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Given an undirected graph G = (V, E), the minimum sum coloring problem (MSCP) is to find a legal vertex coloring of G, using colors represented by natural numbers (1, 2, . . .) such that the total sum of the colors assigned to the vertices is minimized. In this paper, we present EXSCOL, a heuristic algorithm based on independent set extraction for this NP-hard problem. EXSCOL identifies iteratively collections of disjoint independent sets of equal size and assign to each independent set the smallest available color. For the purpose of computing large independent sets, EXSCOL employs a tabu search based heuristic. Experimental evaluations on a collection of 52 DIMACS and COLOR2 benchmark graphs show that the proposed approach achieves highly competitive results. For more than half of the graphs used in the literature, our approach improves the current best known upper bounds.

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Section: The Exscol Algorithmmentioning

confidence: 99%

“…The minimum sum coloring problem is known to be NP-hard in the general case [14]. In addition to its theoretical significance as a difficult combinatorial problem, the MSCP is notable for its ability to formulate a number of important problems, including those from VLSI design, scheduling and resource allocation [1,19].…”

Section: Introductionmentioning

confidence: 99%

An effective heuristic algorithm for sum coloring of graphs

Hao

2012

Computers & Operations Research

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“…Graph Coloring Problem (GCP) and Graph Chromatic Sum problem (GCS) belong to the class of NP-hard combinatorial optimizations problems [6,12].…”

Section: Introductionmentioning

confidence: 99%

On Sum Coloring of Graphs with Parallel Genetic Algorithms

Kokosiński

Kwarciany

Adaptive and Natural Computing Algorithms

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Abstract. Chromatic number, chromatic sum and chromatic sum number are important graph coloring characteristics. The paper proves that a parallel metaheuristic like the parallel genetic algorithm (PGA) can be efficiently used for computing approximate sum colorings and finding upper bounds for chromatic sums and chromatic sum numbers for hardto-color graphs. Suboptimal sum coloring with PGA gives usually much closer upper bounds then theoretical formulas known from the literature.

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“…The problem was first studied independently by Supowit (1987) and by Kubicka & Schwenk (1989) (see also Kubicka 1989). Minimum sum coloring is motivated by applications in scheduling and VLSI design (see e.g., Bar-Noy et al 1998 andNicoloso et al 1999).…”

Section: Introductionmentioning

confidence: 99%

The complexity of chromatic strength and chromatic edge strength

Marx

2006

comput. complex.

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Abstract. The sum of a coloring is the sum of the colors assigned to the vertices (assuming that the colors are positive integers). The sum Σ(G) of graph G is the smallest sum that can be achieved by a proper vertex coloring of G. The chromatic strength s(G) of G is the minimum number of colors that is required by a coloring with sum Σ(G). For every k, we determine the complexity of the question "Is s(G) ≤ k?": it is coNP-complete for k = 2 and Θ p 2 -complete for every fixed k ≥ 3. We also study the complexity of the edge coloring version of the problem, with analogous definitions for the edge sum Σ (G) and the chromatic edge strength s (G). We show that for every k ≥ 3, it is Θ p 2 -complete to decide whether s (G) ≤ k holds. As a first step of the proof, we present graphs for every r ≥ 3 with chromatic index r and edge strength r + 1. For some values of r, such graphs were not known before.

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An introduction to chromatic sums

Cited by 104 publications

References 1 publication

An effective heuristic algorithm for sum coloring of graphs

An effective heuristic algorithm for sum coloring of graphs

On Sum Coloring of Graphs with Parallel Genetic Algorithms

The complexity of chromatic strength and chromatic edge strength

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