a tabu search approach for the sum coloring problem

A tabu search approach for the sum coloringproblem

Hend Bouziri

LARODEC Laboratory, ESSEC, University of Tunis. Tunisia,

Mouna Jouini

LARODEC Laboratory, ISG, University of Tunis, Tunisia.

Abstract

The sum coloring problem has many applications in scheduling. It is a variant ofthe vertex coloring problem where the objective is to minimize the sum of colorsused in coloring the vertices. In this paper, we use tabu search to solve the sum col-oring problem. Experiments are performed on instances extracted from the secondDIMACS challenge. Results show significative improvements on some chromaticsum bounds.

Keywords: sum coloring problem, chromatic sum, strength, tabu search.

1 Introduction

The graph coloring problem (GCP) is defined as coloring the vertices of agraph with the minimum number of colors such that the adjacent nodes arecolored differently. This definition can model all kind of conflict resolutionproblems. For this reason, we can find many extensions of the GCP.

In this work we are interested in the sum coloring problem (SCP). Its goalis to minimize the total sum of colors assigned to vertices. This problem wasfirst introduced by Kubicka [6]. The sum coloring problem can model severalinteresting practical problems in scheduling and ressource allocation [11] [1].

In [7], it was shown that computing the minimum∑

(G) is NP-hard. Apolynomial time algorithm was given for the case where the graph G is atree in [9]. Hence, we have to use approximative approaches to deal with the

Electronic Notes in Discrete Mathematics 36 (2010) 915–922

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sum coloring problem. In the literature, we can note basically the work ofLi et al.[10] on greedy algorithms. Concerning meta heuristics, although alot of work has been reported to deal with the GCP, only a parallel geneticalgorithm was proposed in [5] for the SCP. In this work, we propose a tabusearch approach.

The idea behind the tabu search metaheuristic is to look for a global solu-tion by performing moves in the search space and avoiding stagnation in localsolutions. This approach proved its performance when dealing with severalcombinatorial optimization problems, especially, the graph coloring problem.

This paper reports in section 2 some definitions related to the classicalgraph coloring problem and the sum coloring problem. Section 3 gives theprinciple of the tabu search method and shows the use of this metaheuristicto solve the sum coloring problem. We provide in section 4 results on someinstances extracted from the second DIMACS benchmark.

2 Preliminary definitions and properties

Since the minimum sum coloring problem is a variant of the basic graph color-ing problem (GCP), we start our presentation by defining the graph coloringproblem.

2.1 Graph coloring problem

Given an undirected connected graph G = (V, E), V corresponds to the set ofvertices and E corresponds to the set of edges. The coloring of G which assignsk colors to the vertices of the graph is termed k-coloring. Thus a k-coloringpartitions the vertices of a graph into k color classes. A proper coloring ofa graph G, is an assignment of colors to its vertices so that no two adjacentvertices in G have the same color.

If two adjacent vertices v and w have the same color, then vertices v andw are called conflicting vertices and the edge linking v with w is called aconflicting edge. The chromatic number of a graph G is the minimum numberχ for which G has a χ-coloring. Finding this χ-coloring is called the graphcoloring problem. However, looking for a proper coloring using a given positiveinteger k is called the k-coloring problem.

2.2 Sum coloring problem

The sum coloring problem asks to find a proper coloring of an undirectedgraph G, so that the total sum of colors of vertices is minimised.

H. Bouziri, M. Jouini / Electronic Notes in Discrete Mathematics 36 (2010) 915–922916

Let G be an undirected graph and c a proper coloring of G. The minimumsum

∑(G) is called the chromatic sum, it is denoted by

∑(G) and it is given

by: ∑(G) = minc

∑sum(G, c)

with

sum(G, c) =∑

v∈V (G)

c(v)

The number of colors needed in any coloring with the sum of colors∑

(G)is called the strength of the graph and denoted by s(G). We might thinkthat we can obtain an optimum sum by finding a proper coloring with χ(G).However, the strength of G can be greater than its chromatic number. Forexample, if we consider the tree of the figure 1, we know that χ(G) = 2,however the optimal sum for this tree is equal to 3 colors. Hence,

∑(T ) = 11

and s(T ) = 3.

1 1

1

2 31

1

1

Fig. 1. A graph G with s(G) greater than χ(G).

2.3 Some properties

In this section, we present some theoritical bounds. These results are basicallyextracted from [11] and [8].

Given a proper coloring c of G, we denote by k the number of colors usedin this coloring. Consider the partition of the vertex set induced by coloringc, i.e. independent sets Ci where i = 1..k, if the sum of colors in c provides∑

(G), c is called the best coloring.

For any graph G, the following properties hold:

• |C1| ≥ |C2| ≥ ... ≥ |Ck|,• n ≤ ∑

(G) ≤ m + n,

•√

8|E| ≤ ∑(G) ≤ �3/2(m + 1)�,

• s(G) ≤ �(χ(G)+Δ(G))/2�, where Δ(G) corresponds to the maximal vertexdegree in the graph G,

• s(G).(s(G) + 1)/2 ≤ ∑(G) ≤ n.(χ(G) + 1)/2.

H. Bouziri, M. Jouini / Electronic Notes in Discrete Mathematics 36 (2010) 915–922 917

3 Greedy algorithms for the sum coloring problem

The general idea of a greedy algorithm is to build up a solution, one elementat a time, by choosing the best possible element at each iteration. To dealwith the graph coloring problem, the algorithm starts with the creation of apermutation of vertices, then the first vertex in the permutation is coloredwith the smallest color from {1, .., k + 1}, such that no conflict is created. kis the last color used and in the first step it is equal to 0. Once a vertex iscolored, it is removed from the set of vertices. These two steps are repeatedwith the remaining set of uncolored vertices. Several different schemes havebeen used for the initial ordering.

Two basic greedy algorithms are well used for the graph coloring problem,DSATUR and RLF. The main reason of their effectiveness is that, when se-lecting nodes, they use information from the graph coloring structure. Thisinformation can be the saturation degree or the maximum clique in the graph.In [10], these greedy methods are adapted to deal with the sum coloring prob-lem . Results show experimental upper bounds of the chromatic sum and evenimprove previous solutions obtained for the standard graph coloring problem.

4 Tabu search for the sum coloring problem

Metaheuristics are widely used to solve various combinatorial optimizationproblems. The main reason for the interest in these techniques is their gener-icity. Indeed, a metaheuristic constitutes a general framework which can beapplied to different optimization problems and thus present the ability to beadapted to a specific problem. Generally speaking, metaheuristics use moveoperators to “navigate” from a solution to another one in a search space andthey stop if a satisfactory solution is found or if a stopping criterion is satis-fied. For the sum coloring problem, the use of a metaheuristic might improveprevious results found theoretically or by greedy algorithms. The idea is toexplore a space of k-colorings (which can be non proper) to reach a properk-coloring with a minimum sum of colors.

The tabu search algorithm is considered as one of the most performingmetaheuristics (not hybrids) in the resolution of the graph coloring problem[2]. This insights to use it in the resolution of the sum coloring problem.


4.1 Tabu search principle

The principle of tabu search is to explore iteratively a set of neighboringsolutions of a given solution x and to move towards the best solution x′. Ateach iteration a best neighbor is chosen to replace the current solution. Thetabu search is principally characterized by memory-based strategies. Thismemory allows the use of past experiences to improve the current choice ofsolution. Tabu search uses two kinds of memory:

• Short term memory which corresponds to a tabu list structure. It containsa list of temporary forbidden moves to avoid cycling (visiting a solutionalready explored in the search space).

• Long term memory which can be performed by the use of an aspirationcriterion. It allows a tabu move to be performed even if it is classified astabu.

We can summarize the tabu search procedure in the following algorithm.

Algorithm 1 Tabu search procedure;

Input: an initial solution S0;Output: a local solution local S;BeginSC ←− S0;local S ←− S0;Repeat until a stopping criterion is satisfied;

SV ←−best neighbor of SC obtained with no tabu moveor verifying the aspiration criterion;

SC ←− SV ;

If (fitness(SC) ≤ fitness(local S)) then local S ←− SV ;

Return (local S);End.

This algorithm is adapted to the sum coloring problem in such a waythat solutions correspond to k-colorings which can be proper or not. Initialsolutions are generated at random. The neighborhood considered in this workis to move a conflicting node from a color class to another one. If the solutionhas no conflict, we choose a node and we change its color randomly, amongthe k colors.

Our goal in the search is to minimize the sum of colors in a set of k-colorings of a given graph. For this purpose, let ck a k-coloring of a graph G


with Ec the set of conflicting edges and the partition Ci, i = 1..k induced bythe coloring ck. We have to minimize the following objective function (withM a high valued integer):

f(ck) = M ∗ |Ec| +k∑

i=1

|Ci|∑

j=1

i × j.

5 Results

In this section, we give experimental results related to our algorithm appliedon some instances extracted from the second DIMACS challenge [14]. Wefocus on DSJCn-d random graphs, where n corresponds to the graph size and0.d corresponds to the probability that two nodes are connected by an edgein the graph.

In Table 1 we report in column 1 to 3, respectively the instance name,the best sum of colors found by greedy algorithms in [10] and the best sumfound by our tabu search implementation. The number of colors k is given inbrackets.

Instances Greedy algorithms Tabu search

DSJC125-1 352(6) 344(6)

DSJC125-5 1141(21) 1103(18)

DSJC125-9 2653(50) 2631(49)

DSJC250-1 1068(10) 1046(10)

DSJC250-5 3658 (34) 3779(33)

DSJC250-9 8942(83) 9198(82)

DSJC500-1 3229(15) 3205(15)

Table 1Experimental Results.

Results show that the use tabu search can efficiently improve upper boundsfor graphs DSJC125-d, DSJC125-1 and DSJC500-1. In addition, the up-per bound of the graph strength is improved for the instances DSJC125-5,DSJC125-9 and DSJC250-1. Furthermore, we can see that the tabu searchis very close to improve both chromatic sum and the strength of the other


instances. This is done with a simple tabu search algorithm which starts withrandom non feasible solutions.

6 Conclusion

The sum coloring problem is a relatively new problem in the combinatorialoptimisation field. The main contribution of this paper is to propose a newmeta heuristic for its resolution, namely the tabu search technique. This wasmotivated by two factors, the first one is that the SCP is classified NP-hardon arbitrary graphs. That is no exact method can solve the problem in apolynomial time. The present work tries to show the behavior of the tabusearch when dealing with some instances of the SCP.

The second factor that motivated us to use the tabu search is that it showedhigh performance when dealing with the classical graph coloring problem. In-deed, our experiments show that even with a very simple tabu search schema,results are very competitive and outperform on some instances. In fact, theproposed tabu search technique operates on solutions that can be feasible ornot and moves blindly on the search space.

This implies that incorporating additional information to the search pro-cess might enhance the results. Also, combining our approach with greedyalgorithms can lead to efficient search strategies.

References

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a tabu search approach for the sum coloring problem

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