Elsevier

Applied Soft Computing

Volume 13, Issue 5, May 2013, Pages 2812-2822
Applied Soft Computing

Characterization of graph properties for improved Pareto fronts using heuristics and EA for bi-objective graph coloring problem

https://doi.org/10.1016/j.asoc.2012.06.021Get rights and content

Abstract

Bi-objective graph coloring problem (BOGCP) is a generalized version in which the number of colors used to color the vertices of a graph and the corresponding penalty which incurs due to coloring the end-points of an edge with the same color are simultaneously minimized. In this paper, we have analyzed the graph density, the interconnection between high degree nodes of a graph, the rank exponent of the standard benchmark input graph instances and observed that the characterization of graph instances affects on the behavioral quality of the solution sets generated by existing heuristics across the entire range of the obtained Pareto fronts. We have used multi-objective evolutionary algorithm (MOEA) to obtain improved quality solution sets with the problem specific knowledge as well as with the embedded heuristics knowledge. To establish this fact for BOGCP, hybridization approach is used to construct recombination operators and mutation operators and it is observed from empirical results that the embedded problem specific knowledge in evolutionary operators helps to improve the quality of solution sets across the entire Pareto front; the nature of problem specific knowledge differentiates the quality of solution sets.

Highlights

► Bi-objective graph coloring problem. ► Characterization of graph instances to choose a heuristic. ► Behavior analysis of heuristics. ► MOEA with hybridization approach to obtain improved Pareto front.

Introduction

Graph coloring problem (a.k.a. GCP) is a well-studied single-objective combinatorial optimization problem where the aim is to minimize the number of colors which is used to color the vertices of a given graph G without allowing the same color to the adjacent vertices. Approximating the chromatic number of a given graph G within range n1−∈ for any ∈>0, where n is the cardinality of vertex set and deciding whether G is k (k  3) colorable or not are two well-studied variant of single-objective GCP which belong to NP-hard and NP-complete class, respectively [1], [2], [3]. In this work, we have considered a bi-objective variant of GCP where the number of colors which is used to color the vertices of a given graph G and the penalty that incurs due to coloring adjacent vertices with the same color, are minimized.

The application areas of single objective graph coloring are timetable scheduling, examination scheduling, register allocation, printed circuit testing, electronic bandwidth allocation, microcode optimization, channel routing, the design of flexible manufacturing systems and others. In reality, it may not always be possible to allow the chromatic number of colors to solve the optimization problems. If the number of allowed color is smaller than the chromatic number, it is obvious that a penalty will occur and the goal of the single objective graph coloring may not be solved. With this practical point of view to find the coloring with minimum penalty if the number of allowed color is smaller than the chromatic number and the solution is acceptable with penalty, we aim to work on this bi-objective version of graph coloring problem.

Evolving heuristic algorithms that give approximate or sub-optimal solutions to the considered problem is a widely used method to solve NP-complete optimization problems [4]. Hence, numerous heuristics exist for single-objective GCP. Transformations can be used by heuristic algorithms to give approximate solutions to other NP-complete optimization problems [4]. Moreover, it is easier and less time-consuming to implement and develop than constructing a new heuristic algorithm from scratch. Thus, we have considered and adapted a few single-objective GCP heuristics such as Largest Degree Ordering (LDO) [5], DSatur/Saturated Degree Ordering (SDO) [6], Smallest Last Ordering (SLO) [7], Iterated Greedy [8], and Incidence Degree Ordering (IDO) [9] into the considered bi-objective variant of GCP. Combining heuristics to avoid the minor weaknesses of individual heuristics is a well-known approach to solve optimization problems. Al-Omari and Sabri [9] suggested two combined heuristics where individual LDO and SDO are modified and combined with IDO and LDO, respectively to produce better solution than individual heuristics for single-objective GCP. Well-combined single-objective heuristics for GCP are adapted for bi-objective GCP and penalty adjusting heuristics (PAHs) [10] are considered in this work.

Competing goals and highly complex large search spaces of multi-objective optimization problems boost the need of multiple compromised solution set, known as Pareto-optimal set, instead of single optimal solution for a single objective/goal; hence, evolutionary approaches overpower the rest of the approaches. The considered variant of bi-objective graph coloring is a combinatorial explosive NP-hard problem. Studies have shown that such problems are difficult to solve using EA alone [11], [12], [13]. Multi-objective problems are much more challenging due to diversity of the converged solutions resulting the Pareto front. We have seen that MOEA alone takes enormous amount of time and even that the obtained solutions may not have good diverse and converged solutions. Thus, we have considered hybridization which is known as embedding problem specific knowledge and/or combining nature inspired search techniques.

Problem specific knowledge is embedded in Penalty based Color Partitioning Crossover (PCPX) and Degree Based Crossover (DBX) [10] operators and it has shown that Pareto Converging Genetic Algorithm (PCGA), a steady-state multi-objective evolutionary algorithm (MOEA), produces superior solution-set across the entire range of Pareto front in comparison with considered heuristics [10]. Han at el. proposed a bi-objective evolutionary algorithm (BEA) [14] for a bi-objective variant of GCP to produce optimal penalty for chromatic number. A few work have done using Multi-Objective Genetic Programming (MOGP) to evolve the hyper-heuristics for the considered bi-objective GCP [15] that produce the comparable solution-sets with the combined DSatur—LDO heuristic.

Characterization of the problem instances is needed to choose a particular algorithm for the individual problem instances with the prediction of performance in terms of the quality of solution, computational time, etc. It is also necessary to understand the behavior of an algorithm in advance level. Choosing a particular method i.e. either descriptive or empirical to characterize the problem at instance level is practically difficult. In this work, we have considered the graph density parameter and adapted a few static methods to characterize graph instances. Depending on the characterization of graphs at instance level, a particular solution method can be chosen a priori with the prediction of comparative solution quality. In this work, we have applied hybridization with EA on a bi-objective variant of GCP and analyzed whether the obtained Pareto front is comparable with the obtained Pareto fronts generated by the adaptation of a few well-known heuristics and by MOEA with a few crossover operators such as Penalty based Color Partitioning Crossover (PCPX) [10], Degree Based Crossover (DBX) [10] for the bi-objective variant of GCP. We have analyzed the nature of solution-sets across the complete range of Pareto fronts over each solution method and explored the change in behavior depending on the type of input graph instances.

Section 2 contains the problem formulation and description which includes the adapted heuristics in our work. The characterization of graph instances to analyze the nature of heuristics is described in Section 3. Section 4 describes the hybridization technique of multi-objective evolutionary algorithm which generates the improved quality of solution sets for the bi-objective graph coloring problem. Next, Section 5 contains the empirical results and comparative analysis of heuristics and MOEA solution sets. We draw conclusions in Section 6.

Section snippets

Problem formulation and description

Graph coloring with rejections is one of the standard bi-objective variants of graph coloring problem [16], [17]. The problem is to select a subset of vertices V′ over a set of vertices V for a given graph G = (V, E), where each vertex v is assigned with a rejection cost and to find a proper coloring to the subgraph of G over V′. The objective of this problem is to minimize the total number of colors used to color V′ and total rejection cost of all other vertices.

In this work, we have considered

Behavior analysis of heuristics

We characterize the graph instances to assess the performance of heuristics which are used to solve the considered variant of bi-objective graph coloring problem. Graph coloring problem can be extended to a variety of optimization or constraint satisfaction problems such as examination time-table scheduling and sudoku problem; thus, it must be applicable to variety of graph instances. The prior knowledge of different input graph instances can help to choose the particular heuristic for the

Hybridization approach

The problem specific knowledge of heuristics can boost the efficacy of multi-objective evolutionary algorithm (MOEA) for that problem. In hybridization technique, the specific knowledge can be applied either before or after each MOEA generation in the form of local search technique or can be embedded into evolutionary operators. In this work, the problem specific knowledge is embedded into the evolutionary operators to generate the quality of solution sets for the bi-objective graph coloring

Results and discussions

Pareto Converging Genetic Algorithm (PCGA) [32] is a Pareto-rank based MOEA which includes a rank-histogram scheme to monitor the performance of EA in each successive generation. We have considered PCGA framework for bi-objective graph coloring problem. In this bi-objective graph coloring problem, we have tested our proposed MPGX crossover operator with different crossover probability. Crossover at each point of vertex color within two individuals depends on the dissimilarity of colors at that

Conclusions

In this work, we have considered a bi-objective variant of graph coloring problem where the number of allowable color to color the vertices of a given graph is less than the chromatic number of colors; the aim is to find the minimum penalty graph coloring for a particular number of colors within the valid range of color. In this work, it is shown that the characterization of graph instances helps to choose a particular heuristic for particular application field of bi-objective graph coloring

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