Local search for weighted sum coloring problem

doi:10.1016/j.asoc.2021.107290

Applied Soft Computing

Volume 106, July 2021, 107290

https://doi.org/10.1016/j.asoc.2021.107290 Get rights and content

Highlights

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Adaptive selection for increasing or decreasing the number of color classes.
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Two local search algorithms LS-WSC and CCLS-WSC for weighted sum coloring problem.
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The configuration checking strategy is used in CCLS-WSC algorithm.
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CCLS-WSC algorithm outperforms CPLEX, GREEDY, GA-WSC and LS-WSC on all 142 considered instances.

Abstract

The weighted sum coloring problem (WSCP) is a weighted vertex coloring problem in which a weight is associated with each color. Its aim is to find a coloring of the vertices of a graph with the minimum sum of the costs of the used colors. The WSCP has important applications in the batch scheduling of distributed systems. In this paper, we obtain an integer linear programming (ILP) model for the WSCP, and design an efficient local search algorithm named LS-WSC to solve WSCP. To this end, we propose two ideas to make LS-WSC effective. Firstly, we design an adaptive strategy to change the number of color classes in current solution, which allows LS-WSC to search more feasible solution regions. Secondly, we design a greedy function and a tabu strategy, which make LS-WSC search in right direction. We also design a genetic algorithm GA-WSC to solve WSCP, which is used to experimentally evaluate the result of local search algorithms. The preliminary experimental results show that LS-WSC outperforms the famous CPLEX on almost all instances, and is competitive with the approximate algorithm GREEDY on most benchmark instances with vertex number less than 200. However, the performance of LS-WSC is worse than GREEDY algorithm $^{'}$ s on most benchmark instances with vertex number greater than 200. Overall, the performance of LS-WSC is only marginally better than GA-WSC $^{'}$ s. In order to improve the performance of LS-WSC, we use the configuration checking strategy to update the searching process of LS-WSC, and the new algorithm for solving WSCP is named as CCLS-WSC. The further experimental results show that CCLS-WSC algorithm outperforms the famous CPLEX, the approximate algorithm GREEDY, the genetic algorithm GA-WSC and LS-WSC on all benchmark instances.

Introduction

The very famous NP-hard vertex coloring problem (VCP) is to find a valid coloring while minimizing the number of colors used [1]. Graph coloring problems can also concern weighted graphs where a weight (typically a positive value) is associated to each vertex [2], [3]. The sum coloring problem (SCP) is a variant of the VCP and aims to determine a proper coloring while minimizing the sum of the colors assigned to the vertices [4], [5]. The weighted sum coloring problem (WSCP) is a variant of the SCP in which a weight is associated with each color [6]. For each color in WSCP, its weight is the greatest weight of a vertex in it. Informally, the WSCP aims to find a legal coloring such that the chromatic sum of the weighted graph is minimized. A formal definition of WSCP is given in Section 2.1 together with an illustrative example.

It should be noted that an instance of the NP-hard sum coloring problem can be easily reduced to an instance of the WSCP by assigning a weight of 1 for each vertex [7]. As a result, it can be reasonably inferred that the WSCP is also NP-hard. Based on the computational challenge of WSCP and the importance of graph coloring problem, to study the solution of WSCP is of great theoretical significance. WSCP also has important applications in the batch scheduling of distributed systems, in which jobs have different processing times. There are two application directions for WSCP in batch scheduling. One is the batch scheduling of conflicting jobs, in which some jobs use the same resources and then cannot be processed concurrently [6], [8]. And the other one is batch scheduling of job processing time compatibilities, in which only compatible jobs can be processed simultaneously [9], [10], [11]. As a result, effective solution methods for the WSCP can help to solve these practical problems.

Despite the theoretical and practical significance of the WSCP, our literature review given in Section 2.2 indicates that unlike the popular vertex coloring problem, such as the weighted vertex coloring problem and the sum coloring problem, there are few heuristics or exact algorithms which help to satisfy the good performance in real-world applications of WSCP. In this work, we aim to fill the gap by designing effective heuristics that find high-quality approximate solutions for problem instances.

Our work on designing efficient local search algorithms for WSCP is driven by the following consideration. The minimum completion time problem of batch scheduling with different job weights can be naturally encoded as WSCP, which requires the good performance of WSCP solvers. However, the existing algorithms for solving WSCP are inefficient and not applicable [6], [12]. A number of heuristic algorithms are applied into WVCP and SCP [13], [14], [15], [16], [17], [18], which can be used to find high-quality approximate solutions for problem instances that cannot be solved exactly. Based on previous studies of heuristic algorithms for solving other related coloring problems, we present the first study of local search methods for solving WSCP in this work. We summarize the main contributions of this work as follows.

First, two efficient local search algorithm named LS-WSC and CCLS-WSC are proposed in this paper. LS-WSC is designed based on two new ideas. One of them is that an adaptive strategy to changing the number of color classes of current solution is proposed. In each searching step, LS-WSC changes the number of color classes, only when the weight of the random vertex is greatest in the color class which contains it. The other one is that a greedy function and a tabu strategy are designed for LS-WSC. The greedy function makes sure that LS-WSC can search better solutions in current solution region, and the tabu strategy helps to ensure that LS-WSC does not search the recently searched solution region. CCLS-WSC algorithm is designed by using the famous configuration checking strategy to control the searching of LS-WSC, which is used to further improve the performance of LS-WSC.

Second, we assess the proposed algorithms on 142 conventional benchmark instances from the literature (two sets of 77 instances from the DIMACS/COLOR competitions and two sets of 65 instances from matrix-decomposition problems). The best solutions of the above 142 benchmarks found by five compared algorithms (LS-WSC, CCLS-WSC, the genetic algorithm GA-WSC, the approximate algorithm GREEDY and the commercial solver CPLEX) are presented in this paper. These results and the two proposed algorithms LS-WSC and CCLS-WSC can be used to assess future WSCP algorithms, and these two new local search algorithms can also serve as upper bound methods to design effective exact algorithms.

The rest sections of this paper are organized as follows. The problem definition and literature review of WSCP are described in Section 2. The preliminaries of local search algorithm and ILP model of WSCP are described in Section 3. Section 4 introduces the proposed LS-WSC algorithm, the greedy initialization process and tabu search process in detail. Section 5 presents the experimental results of LS-WSC and comparisons with the genetic algorithm GA-WSC, the commercial solver CPLEX and the approximate algorithm GREEDY on 142 classical instances. The CCLS-WSC and further experiments are presented in Section 6, followed by the discussion of our conclusions in Section 7.

Section snippets

Weighted sum coloring problem

Consider an undirected, weighted graph $G$ $=$ ( $V$ , $E$ , $W$ ), where $V$ denotes the set of vertices, $E$ denotes the set of edges, and $W$ is a non-negative weight function $V$ $\to$ $R^{+}$ associated with vertices. An independent set (a stable or a color class) of $G$ is a subset $V_{i}$ of $V$ such that $\forall u$ , $v$ $\in$ $V_{i}$ , (u,v) $\notin E$ . A feasible $k$ -coloring of $G$ can be defined as a partition of the vertex set $V$ into $k$ disjoint independent sets { $V_{1}$ , $V_{2}$ ,…, $V_{k}$ }. We shall use $W$ ( $V_{i}$ ) to represent the weight of stable $V_{i}$ , which is the

Preliminaries

Consider an undirected, weighted graph $G$ $=$ ( $V$ , $E$ , $W$ ), where $V$ denotes the set of vertices, $E$ denotes the set of edges, and $W$ is a non-negative weight function $V$ $\to$ $R^{+}$ associated with vertices. We shall use w $_{v}$ to represent a non-negative weight of each vertex $v$ $\in$ $V$ . We shall use $N$ ( $v$ ) to represent the neighbor vertex set of $v$ . We also denote $N$ [ $v$ ] $=$ $N$ ( $v$ ) $\cup$ $v$ . We shall use $C$ ( $v$ ) to represent the color class which includes $v$ .

In order to guide the search toward the direction with more

An efficient local search algorithm for the WSCP

We introduce an efficient local search algorithm LS-WSC for the WSCP in this section. The LS-WSC consists of a greedy initialization phase and a tabu search phase. We first introduce the general approach and then explain the components of the proposed algorithm including the greedy initialization, reducing number of color classes and searching a better solution.

Experimental results of LS-WSC algorithm

In this section, we perform intensive experiments to evaluate the proposed LS-WSC for solving WSCP. We compare LS-WSC with the famous CPLEX, the genetic algorithm GA-WSC for solving WSCP and an approximate algorithm named GREEDY [6] on 142 benchmark instances.

Searching a better solution based on configuration checking

In order to further improve the performance of LS-WSC, we adopt an effective strategy named configuration checking to control the selection of vertices in the procedure of searching a better solution. The configuration checking strategy was firstly proposed by Cai [36] and has been already used in many problems, including maximum k-plex problem [37], boolean satisfiability [38], and set covering [39].

The basic principle of configuration checking is that only changing the statuses of vertices

Conclusions and future work

In this paper, we developed two new local search algorithms for WSCP problem called LS-WSC and CCLS-WSC, which work particularly well for 142 classical instances. Two strategies are proposed to improve the local search process of LS-WSC in this paper. First, an adaptive selection strategy for changing the number of color classes is used in LS-WSC. Second, a greedy function and a tabu strategy are used to correct the searching process of LS-WSC. CCLS-WSC updates LS-WSC by improving the procedure

Funding

This work is supported by the Natural Science Basic Research Program of Shaanxi ,China (under Grant No. 2020JQ-280), Shaanxi’s Key Research and Development Program ,China (under Grand No. 2019ZDLNY07-06-01), Research Fund of Guangxi Key Lab of Multi-source Information Mining & Security ,China (under Grant No. MIMS19-05), National Natural Science Foundation of China (under Grant No. 61976050), Jilin Provincial Science and Technology Department ,China (under Grant No. 20190302109GX).

CRediT authorship contribution statement

Dangdang Niu: Methodology, Proponents of major academic ideas, Supervision, Writing - original draft. Bin Liu: Data processing, Figure plotting, Writing - original draft. Minghao Yin: Polishing the English presentation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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View full text

Local search for weighted sum coloring problem

Highlights

Abstract

Introduction

Section snippets

Weighted sum coloring problem

Preliminaries

An efficient local search algorithm for the WSCP

Experimental results of LS-WSC algorithm

Searching a better solution based on configuration checking

Conclusions and future work

Funding

CRediT authorship contribution statement

Declaration of Competing Interest

Discrete Optim.

Theoret. Comput. Sci.

Theoret. Comput. Sci.

Discrete Appl. Math.

Electron. Notes Discrete Math.

Inform. Sci.

Appl. Soft Comput.

Electron. Notes Discrete Math.

Comput. Oper. Res.

Electron. Notes Discrete Math.

Electron. Notes Discrete Math.

Inf. Sci.

European J. Oper. Res.

Artificial Intelligence

A Guide to Graph Colouring - Algorithms and Applications

The vertex coloring problem and its generalizations

4OR: Q. J. Belg., Fr. Italian Oper. Res. Soc.

The Chromatic Sum of a Graph

Algorithms for the minimum sum coloring problem: a review

Artif. Intell. Rev.

Weighted sum coloring in batch scheduling of conflicting jobs

Algorithmica