Ceng RQ(曾荣强)

Hu ZY(胡正银)

Li-Yuan Xue

Wen Y(文奕)

English

National Science Library,Chinese Academy of Sciences

Solving the Bi-criteria Max-Cut Problem withDifferent Neighborhood Combination StrategiesLi-Yuan Xue1, Rong-Qiang Zeng2,3(B), Zheng-Yin Hu3, and Yi Wen31 EHF Key Laboratory of Science, School of Electronic Engineering,University of Electronic Science and Technology of China,Chengdu 611731, Sichuan, People’s Republic of Chinaxuely2013@gmail.com2 School of Mathematics, Southwest Jiaotong University,Chengdu 610031, Sichuan, People’s Republic of Chinazrq@swjtu.edu.cn3 Chengdu Documentation and Information Center, Chinese Academyof Sciences, Chengdu 610041, Sichuan, People’s Republic of China{huzy,wenyi}@clas.ac.cnAbstract. Local search is known to be a highly effective metaheuristicframework for solving a number of classical combinatorial optimizationproblems, which strongly depends on the characteristics of neighborhoodstructure. In this paper, we integrate different neighborhood combina-tion strategies into the hypervolume-based multi-objective local searchalgorithm, in order to deal with the bi-criteria max-cut problem. Theexperimental results indicate that certain combinations are superior toothers and the performance analysis sheds lights on the ways to furtherimprovements.Keywords: Multi-objective optimization · Hypervolume contribution ·Neighborhood combination · Local search · Max-cut problem1 IntroductionLocal search is a simple and effective metaheuristic framework for solving anumber of classical combinatorial optimization problems, which proceeds froman initial solution with a sequence of local changes by defining the proper neigh-borhood structure for the considered problem. In order to study different neigh-borhood combination strategies during the local search process, we present theexperimental analysis of different neighborhoods to solve the bi-criteria max-cutproblem.Given an undirected graph G = (V,E) with the vertex set V = {1, . . . , n}and the edge set E ⊂ V × V . Each edge (i, j) ∈ E is associated with a weightwij . The max-cut problem is to seek a partition of the vertex set V into twodisjoint subsets V1 and V2, which is mathematically formulated as follows [1]:fk(V1, V2) = max∑i∈V1,j∈V2wkij (1)c© Springer International Publishing AG 2017H. Yin et al. (Eds.): IDEAL 2017, LNCS 10585, pp. 508–515, 2017.https://doi.org/10.1007/978-3-319-68935-7 55Neighborhood Combination Strategies 509where wkij is the weight of the kth (k ∈ {1, 2}) graph. As one of Karps 21NP-complete problems with numerous practical applications [4], a large numberof metaheuristics have been proposed to tackle this problem, including scattersearch [6], global equilibrium search [7], tabu search [8], etc.In this paper, we integrate different neighborhood combination strategies intothe hypervolume-based multi-objective local search algorithm, in order to studythe search capability of different neighborhood combinations on the bi-criteriamax-cut problem. The experimental results indicate that certain combinationsare superior to others. The performance analysis explains the behavior of thealgorithms and sheds lights on the ways to further enhance the search.The remaining part of this paper is organized as follows. In the next section,we briefly introduce the basic notations and definitions of bi-objective optimiza-tion. In Sect. 3, we present the hypervolume-based multi-objective local searchalgorithm with different neighborhood combination strategies for solving bi-criteria max-cut problem. Section 4 indicates that the experimental results onthe benchmark instances of max-cut problem. The conclusions are provided inthe last section.2 Bi-objective OptimizationIn this section, we briefly introduce the basic notations and definitions of bi-objective optimization. Without loss of generality, we assume that X denotesthe search space of the optimization problem under consideration and Z = 2denotes the corresponding objective space with a maximizing vector functionZ = f(X), which defines the evaluation of a solution x ∈ X [5]. Specifically, thedominance relations between two solutions x1 and x2 are presented below [9]:Definition 1 (Pareto Dominance). A decision vector x1 is said to dominateanother decision vector x2 (written as x1  x2), if fi(x1) ≥ fi(x2) for all i ∈{1, 2} and fj(x1) > fj(x2) for at least one j ∈ {1, 2}.Definition 2 (Pareto Optimal Solution). x ∈ X is said to be Pareto optimal ifand only if there does not exist another solution x′ ∈ X such that x′  x.Definition 3 (Non-Dominated Solution). x ∈ S (S ⊂ X) is said to be non-dominated if and only if there does not exist another solution x′ ∈ S such thatx′  x.Definition 4 (Pareto Optimal Set). S is said to be a Pareto optimal set if andonly if S is composed of all the Pareto optimal solutions.Definition 5 (Non-dominated Set). S is said to be a non-dominated set if andonly if any two solutions x1 ∈ S and x2 ∈ S such that x1  x2 and x2  x1.Actually, we are interested in finding the Pareto optimal set, which keepsthe best compromise among all the objectives. However, it is very difficult oreven impossible to generate the Pareto optimal set in a reasonable time for theNP-hard problems. Therefore, we aim to obtain a non-dominated set which isas close to the Pareto optimal set as possible. That’s to say, the whole goal is toidentify a Pareto approximation set with high quality.510 L.-Y. Xue et al.3 Neighborhood Combination StrategiesIn this work, we integrate different neighborhood combination strategies intothe hypervolume-based multi-objective local search algorithm, in order to solvebi-criteria max-cut problem. The general scheme of Hypervolume-Based Multi-Objective Local Search (HBMOLS) algorithm [3] is presented in Algorithm 1,and the main components of this algorithm are described in detail in the followingsubsections.Algorithm 1. Hypervolume-Based Multi-Objective Local Search AlgorithmInput: N (Population size)Output: A: (Pareto approximation set)Step 1 - Initialization: P ← N randomly generated solutionsStep 2: A ← ΦStep 3 - Fitness Assignment: Assign a fitness value to each solution x ∈ PStep 4:while Running time is not reached dorepeatHypervolume-Based Local Search: x ∈ Puntil all neighbors of x ∈ P are exploredA ← Non-dominated solutions of A⋃Pend whileStep 5: Return AIn HBMOLS, each individual in an initial population is generated by ran-domly assigning the vertices of the graph to the two vertex subsets V1 and V2.Then, we employ a Hypervolume Contribution (HC) indicator proposed in [3]to achieve the fitness assignment for each individual. Based on the dominancerelation and two objective function values, the HC indicator calculate the hyper-volume contribution of each individual in the objective space.For the hypervolume-based local search procedure, we implement the f -flip(f ∈ {1, 2}) move based neighborhood strategy with the combinations. After-wards, each solution is optimized by the hypervolume-based local search proce-dure, which will repeat until the termination criterion is satisfied, so as to obtaina high-quality Pareto approximation set.3.1 One-Flip MoveIn order to deal with the max-cut problem, one-flip move is realized by movinga randomly selected vertex to the opposite set, which is calculated as follows:Δi =∑x∈V1,x =v1wvix −∑y∈V2wviy, vi ∈ V1 (2)Δi =∑x∈V2,y =v1wviy −∑y∈V1wvix, vi ∈ V2 (3)Neighborhood Combination Strategies 511Let Δi be the move gain of representing the change in the fitness function,and Δi can be calculated in linear time by the formula above, more details aboutthis formula can be found in [8]. Then, we can calculate the objective functionvalues high efficiently with the streamlined incremental technique.3.2 Two-Flip MoveIn the case of two-flip move, we can obtain a new neighbor solution by randomlymoving two different vertices vi and vj from the set V1 to another set V2. Infact, two-flip move can be seen as a combination of two single one-flip moves.We denote the move value by δij , which is derived from two one-flip moves Δiand Δj (i = j) as follows:δij = Δi + Δj (4)Especially, the search space generated by two-flip move is much bigger than theone generated by one-flip move. In the following, we denote the neighborhoodswith one-flip move and two-flip move as N1 and N2 respectively.4 ExperimentsIn this section, we present the experimental results of 3 different neighborhoodcombination strategies on 9 groups of benchmark instances of max-cut problem.All the algorithms are programmed in C++ and compiled using Dev-C++ 5.0compiler on a PC running Windows 7 with Core 2.50 GHz CPU and 4GB RAM.4.1 Parameters SettingsIn order to conduct the experiments on the bi-objective max-cut problem, weuse two single-objective benchmark instances of max-cut problem with thesame dimension provided in [4]1 to generate one bi-objective max-cut probleminstance. All the instances used for experiments are presented in Table 1 below.In addition, the algorithms need to set a few parameters, we only discuss twoimportant ones: the running time and the population size, more details aboutthe parameter settings for multi-objective optimization algorithms can be foundin [2,8]. The exact information about the parameter settings in our work ispresented in the following Table 2.4.2 Performance Assessment ProtocolIn this paper, we evaluate the efficacy of 3 different neighborhood combinationstrategies with the performance assessment package provided by Zitzler et al.2.1 More information about the benchmark instances of max-cut problem can be foundon this website: http://www.stanford.edu/∼yyye/yyye/Gset/.2 More information about the performance assessment package can be found on thiswebsite: http://www.tik.ee.ethz.ch/pisa/assessment.html.512 L.-Y. Xue et al.Table 1. Single-objective benchmark instances of max-cut problem used for generatingbi-objective max-cut problem instances.Dimension Instance 1 Instance 2bo mcp 800 01 800 g1.rud g2.rudbo mcp 800 02 800 g11.rud g12.rudbo mcp 800 03 800 g15.rud g19.rudbo mcp 800 04 800 g17.rud g21.rudbo mcp 2000 01 2000 g22.rud g23.rudbo mcp 2000 02 2000 g32.rud g33.rudbo mcp 2000 03 2000 g35.rud g39.rudbo mcp 1000 01 1000 g43.rud g44.rudbo mcp 3000 01 3000 g49.rud g50.rudTable 2. Parameter settings used for bi-objective max-cut problem instances: instancedimension (D), vertices (V ), edges(E), population size (P ) and running time (T ).Dimension (D) Vertices (V ) Edges (E) Population (P ) Time (T )bo mcp 800 01 800 800 19176 20 40′′bo mcp 800 02 800 800 1600 20 40′′bo mcp 800 03 800 800 4661 20 40′′bo mcp 800 04 800 800 4667 20 40′′bo mcp 2000 01 2000 2000 19990 50 100′′bo mcp 2000 02 2000 2000 4000 50 100′′bo mcp 2000 03 2000 2000 11778 50 100′′bo mcp 1000 01 1000 1000 9990 25 50′′bo mcp 3000 01 3000 3000 6000 75 150′′The quality assessment protocol works as follows: First, we create a set of 20 runswith different initial populations for each strategy and each benchmark instanceof max-cut problem. Then, we generate the reference set RS∗ based on the 60different sets A0, . . . , A59 of non-dominated solutions.According to two objective function values, we define a reference point z =[r1, r2], where r1 and r2 represent the worst values for each objective functionin the reference set RS∗. Afterwards, we assign a fitness value to each non-dominated set Ai by calculating the hypervolume difference between Ai andRS∗. Actually, this hypervolume difference between these two sets should be asclose to zero as possible [10].Neighborhood Combination Strategies 5134.3 Computational ResultsIn this subsection, we present the computational results on 9 groups of bi-objective max-cut problem instances, which are obtained by three differentneighborhood combination strategies. The information about these algorithmsare described in the following table:Table 3. The algorithms with different neighborhood combination strategies.Algorithm descriptionHBMOLS N1 One-flip move based local searchHBMOLS N2 Two-flip move based local searchHBMOLS (N1⋃N2) f -flip move based local search (f ∈ {1, 2})In Table 3, the algorithms HBMOLS (N1⋃N2) selects one of the two neigh-borhoods to be implemented at each iteration during the local search process,choosing the neighborhood N1 with a predefined probability p and choosing N2with the probability 1 − p. In our experiments, we set the probability p = 0.5.The computational results are summarized in Table 4. In this table, there isa value both in bold and in grey box at each line, which is the best resultobtained on the considered instance. The values both in italic and bold ateach line refer to the corresponding algorithms which are not statistically out-performed by the algorithm obtaining the best result (with a confidence levelgreater than 95%).Table 4. The computational results on bi-criteria max-cut problem obtained by thealgorithms with 4 different neighborhood combination strategies.514 L.-Y. Xue et al.From Table 4, we can observe that all the best results are obtained by N1,which statistically outperforms the other two algorithms on all the instances.Moreover, the results obtain by N1⋃N2 is close to the results obtainedby N1. Especially, the most significant result is achieved on the instancebo mcp 2000 01, where the average hypervolume difference value obtained byN1 is much smaller than the values obtained by the other two algorithms.Nevertheless, N2 does not perform as well as N1, although the search spaceof them is much bigger than N1. We suppose that there exists some key ver-tices in the representation of the individuals, which means these vertices shouldbe assigned in some set in order to search the local optima effectively. Two-flip moves change the positions of the key vertices much more frequently thanthe one-flip move, then the efficiency of local search is obviously affected bythis neighborhood strategy. Actually, N1⋃N2 provides a possibility to keep thepositions of the key vertices unchanged and broaden the search space. Thus, thecombination of one-flip move and two-flip move is very potential to obtain betterresults.5 ConclusionIn this paper, we have presented different neighborhood combination strategiesto deal with the bi-criteria max-cut problem, which are based on one-flip, two-flipand the combination. For this purpose, we have carried out the experiments on9 groups of benchmark instances of max-cut problem. The experimental resultsindicate that the better outcomes are achieved with the simple one-flip movebased neighborhood and the neighborhood combination with two-flip is verypotential to escape the local optima for further improvements.Acknowledgments. The work in this paper was supported by the FundamentalResearch Funds for the Central Universities (Grant No. A0920502051722-53) and sup-ported by the West Light Foundation of Chinese Academy of Science (Grant No:Y4C0011001).References1. Angel, E., Gourves, E.: Approximation algorithms for the bi-criteria weighted max-cut problem. Discrete Appl. Math. 154, 1685–1692 (2006)2. Basseur, M., Liefooghe, A., Le, K., Burke, E.: The efficiency of indicator-basedlocal search for multi-objective combinatorial optimisation problems. J. Heuristics18(2), 263–296 (2012)3. Basseur, M., Zeng, R.-Q., Hao, J.-K.: Hypervolume-based multi-objective localsearch. Neural Comput. Appl. 21(8), 1917–1929 (2012)4. Benlic, U., Hao, J.-K.: Breakout local search for the max-cut problem. Eng. Appl.Artif. Intell. 26, 1162–1173 (2013)5. Coello, C.A., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithmsfor Solving Multi-Objective Problems (Genetic and Evolutionary Computation).Springer, Secaucus (2007)Neighborhood Combination Strategies 5156. Marti, R., Duarte, A., Laguna, M.: Advanced scatter search for the max-cut prob-lem. INFORMS J. Comput. 21(1), 26–38 (2009)7. Shylo, V.P., Shylo, O.V.: Solving the maxcut problem by the global equilibriumsearch. Cybern. Syst. Anal. 46(5), 744–754 (2010)8. Wu, Q., Wang, Y., Lu¨, Z.: A tabu search based hybrid evolutionary algorithm forthe max-cut problem. Appl. Soft Comput. 34, 827–837 (2015)9. Zitzler, E., Ku¨nzli, S.: Indicator-based selection in multiobjective search. In: Yao,X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervo´s, J.J., Bullinaria, J.A.,Rowe, J.E., Tinˇo, P., Kaba´n, A., Schwefel, H.-P. (eds.) PPSN 2004. LNCS, vol.3242, pp. 832–842. Springer, Heidelberg (2004). doi:10.1007/978-3-540-30217-9 8410. 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Solving the Bi-criteria Max-Cut Problem with Different Neighborhood Combination Strategies

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Solving the Bi-criteria Max-Cut Problem with Different Neighborhood Combination Strategies

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