A comparative analysis of two matheuristics by means of merged local optima networks

Abstract We present a comparative analysis of two hybrid algorithms for solving combinatorial optimisation problems. The first one is a specific variant of an established family of techniques known as large neighbourhood search (LNS). The second one is a much more recent algorithm known as construct, merge, solve & adapt (CMSA). Both approaches generate, in different ways, reduced sub-instances of the tackled problem instance at each iteration. The experimental analysis is conducted on two NP-hard combinatorial subset selection problems: the multidimensional knapsack problem and minimum common string partition. The results support the intuition that CMSA has advantages over the LNS variant in the context of problems for which solutions contain rather few items. Moreover, they show that the opposite may be the case for problems in which solutions contain rather many items. The analysis is supported by a new way of visualising the trajectories of the compared algorithms in terms of merged monotonic local optima networks.

[1]  Dan He,et al.  A Novel Greedy Algorithm for the Minimum Common String Partition Problem , 2007, ISBRA.

[2]  Plácido Rogério Pinheiro,et al.  Tackling the Container Loading Problem: A Hybrid Approach Based on Integer Linear Programming and Genetic Algorithms , 2007, EvoCOP.

[3]  Christian Blum,et al.  Hybrid Metaheuristics , 2019, Lecture Notes in Computer Science.

[4]  Tao Jiang,et al.  Computing the Assignment of Orthologous Genes via Genome Rearrangement , 2005, APBC.

[5]  Sébastien Vérel,et al.  Local Optima Networks of NK Landscapes With Neutrality , 2011, IEEE Transactions on Evolutionary Computation.

[6]  Edward M. Reingold,et al.  Tidier Drawings of Trees , 1981, IEEE Transactions on Software Engineering.

[7]  Christian Blum,et al.  Hybrid Metaheuristics: Powerful Tools for Optimization , 2016 .

[8]  Verena Schmid,et al.  Hybrid large neighborhood search for the bus rapid transit route design problem , 2014, Eur. J. Oper. Res..

[9]  Maria Grazia Speranza,et al.  Kernel search: A general heuristic for the multi-dimensional knapsack problem , 2010, Comput. Oper. Res..

[10]  Edward M. Reingold,et al.  Graph drawing by force‐directed placement , 1991, Softw. Pract. Exp..

[11]  Stephen C. H. Leung,et al.  A hybrid simulated annealing metaheuristic algorithm for the two-dimensional knapsack packing problem , 2012, Comput. Oper. Res..

[12]  Fred W. Glover,et al.  A two-phase tabu-evolutionary algorithm for the 0-1 multidimensional knapsack problem , 2018, Inf. Sci..

[13]  John E. Beasley,et al.  A Genetic Algorithm for the Multidimensional Knapsack Problem , 1998, J. Heuristics.

[14]  Dong Yue,et al.  Diversity-preserving quantum particle swarm optimization for the multidimensional knapsack problem , 2020, Expert Syst. Appl..

[15]  Saïd Hanafi,et al.  An efficient tabu search approach for the 0-1 multidimensional knapsack problem , 1998, Eur. J. Oper. Res..

[16]  Pierre Hansen,et al.  Variable neighborhood search: Principles and applications , 1998, Eur. J. Oper. Res..

[17]  Kirk Pruhs,et al.  Approximation schemes for a class of subset selection problems , 2007, Theor. Comput. Sci..

[18]  Paul D. Seymour,et al.  Tour Merging via Branch-Decomposition , 2003, INFORMS J. Comput..

[19]  Steven Li,et al.  Solving large-scale multidimensional knapsack problems with a new binary harmony search algorithm , 2015, Comput. Oper. Res..

[20]  Gilbert Laporte,et al.  An adaptive large neighborhood search heuristic for the Pollution-Routing Problem , 2012, Eur. J. Oper. Res..

[21]  David Pisinger,et al.  Large Neighborhood Search , 2018, Handbook of Metaheuristics.

[22]  G. Dueck,et al.  Record Breaking Optimization Results Using the Ruin and Recreate Principle , 2000 .

[23]  Gábor Csárdi,et al.  The igraph software package for complex network research , 2006 .

[24]  Sébastien Vérel,et al.  A study of NK landscapes' basins and local optima networks , 2008, GECCO '08.

[25]  Marco Tomassini,et al.  Understanding Phase Transitions with Local Optima Networks: Number Partitioning as a Case Study , 2017, EvoCOP.

[26]  Antonio Bolufé Röhler,et al.  Matheuristics: Optimization, Simulation and Control , 2009, Hybrid Metaheuristics.

[27]  Stefan Voß,et al.  POPMUSIC as a matheuristic for the berth allocation problem , 2014, Annals of Mathematics and Artificial Intelligence.

[28]  Pierre Dejax,et al.  A large neighborhood search heuristic for supply chain network design , 2014, Comput. Oper. Res..

[29]  Christian Blum,et al.  Minimum common string partition: on solving large-scale problem instances , 2018, Int. Trans. Oper. Res..

[30]  Petr Kolman,et al.  Minimum Common String Partition Problem: Hardness and Approximations , 2004, Electron. J. Comb..

[31]  Gabriela Ochoa,et al.  Perturbation Strength and the Global Structure of QAP Fitness Landscapes , 2018, PPSN.

[32]  Abraham P. Punnen,et al.  Very Large-Scale Neighborhood Search , 2000, Handbook of Approximation Algorithms and Metaheuristics.

[33]  Leslie Pérez Cáceres,et al.  The irace package: Iterated racing for automatic algorithm configuration , 2016 .

[34]  Ravindra K. Ahuja,et al.  Very large-scale neighborhood search , 2000 .

[35]  Fred W. Glover,et al.  Ejection Chains, Reference Structures and Alternating Path Methods for Traveling Salesman Problems , 1996, Discret. Appl. Math..

[36]  J. Doye.  Network topology of a potential energy landscape: a static scale-free network. , 2002, Physical review letters.

[37]  Hans Kellerer,et al.  Multidimensional Knapsack Problems , 2004 .

[38]  Christian Blum,et al.  Mathematical programming strategies for solving the minimum common string partition problem , 2014, Eur. J. Oper. Res..

[39]  Plácido Rogério Pinheiro,et al.  A Two-Phase Approach for Single Container Loading with Weakly Heterogeneous Boxes , 2019, Algorithms.

[40]  Gabriela Ochoa,et al.  Mapping the global structure of TSP fitness landscapes , 2017, J. Heuristics.

[41]  Marco Caserta,et al.  A corridor method based hybrid algorithm for redundancy allocation , 2014, Journal of Heuristics.

[42]  William J. Cook,et al.  Finding Tours in the TSP , 1999 .

[43]  José A. Moreno-Pérez,et al.  A Kernel Search Matheuristic to Solve The Discrete Leader-Follower Location Problem , 2019, Networks and Spatial Economics.

[44]  Christian Blum,et al.  Hybrid Metaheuristics , 2010, Artificial Intelligence: Foundations, Theory, and Algorithms.

[45]  Sébastien Vérel,et al.  Local Optima Networks of the Quadratic Assignment Problem , 2010, IEEE Congress on Evolutionary Computation.

[46]  Matteo Fischetti,et al.  Local branching , 2003, Math. Program..

[47]  Maria Grazia Speranza,et al.  Kernel Search: An application to the index tracking problem , 2012, Eur. J. Oper. Res..

[48]  Christian Blum,et al.  Hybrid techniques based on solving reduced problem instances for a longest common subsequence problem , 2018, Appl. Soft Comput..

[49]  Arnaud Fréville,et al.  An Efficient Preprocessing Procedure for the Multidimensional 0- 1 Knapsack Problem , 1994, Discret. Appl. Math..

[50]  Christian Blum,et al.  Construct, Merge, Solve and Adapt Versus Large Neighborhood Search for Solving the Multi-dimensional Knapsack Problem: Which One Works Better When? , 2017, EvoCOP.

[51]  Manuel López-Ibáñez,et al.  Construct , Merge , Solve & Adapt : A New General Algorithm For Combinatorial Optimization , 2015 .