Hybrids combining Local Search Heuristics with Exact Algorithm Susana Fernandes 1, Helena R. Lourenço 2 Abstract-- Recently hybrid metaheuristics have been design to 4) Local search guided by information from find solutions for combinatorial optimization problems. We integer programming relaxations. focus on hybrid procedures that combine local search based metaheuristics with exact algorithms of the operations 5) Use exact algorithms for specific procedures research field. We present a mapping that outlines the within metaheuristics. metaheuristic and exact procedures used, the way they are related and the problems they have been applied to. The next classification done by (Puchinger and Keywords— hybrid metaheuristics, exact algorithms, Raild 2005) considers a combination of exact combinatorial optimization algorithms. methods and metaheuristics and includes the following categories: 1. Collaborative algorithms exchange information I. INTRODUCTION but are not part of each other. The authors consider two subcategories: one, sequential the In this research we are particularly interested in other parallel and intertwined. In sequential hybrid procedures that combine local search based execution, one technique does a preprocessing metaheuristics with exact algorithms of the before the other or the second one is a post operations research field. We name them Optimized processing of the solution(s) generated by the Search Heuristics (OSH). We present how this kind first. Sometimes both techniques have equal of procedures has been applied to combinatorial importance and we cannot speak of pre or post optimization problems. We start by comparing and processing. In parallel execution, several examining the correspondences of two existent processors perform simultaneous tasks acting as classifications of such procedures and transform the teams and interchanging information. In more general one adding a new item and renaming intertwined execution a single processor other. To stress the distribution of these applications executes some steps of one procedure, then over the different problems of combinatorial some steps of another. optimization, we group the problems following a 2. Integrative combinations. One technique is a classification of NP optimization problems and subordinate embedded component of the other indicate the work where OSH methods were applied technique. The authors consider the following to these problems. We also present a large number 2.1. Incorporating exact algorithms in of references that use Optimized search Heuristics. metaheuristics by solving exactly relaxed problems - (as solutions to relaxations II. CLASSIFICATION OF OSH PROCEDURES heuristically guide neighbourhood search, In the literature we can find two classifications recombination, mutation, repair and/or of Optimized Search Heuristics. The first one, done local improvement); or by searching by (Dumitrescu and Stützle 2003), presents a exactly large neighborhoods (exact classification of solution methods that combines algorithms are used to search local search with exact algorithms. In particular they neighborhoods in local search based consider that the main framework is based on local metaheuristics). Merging solutions - Exact search and the subproblems are approached by exact algorithms are used to solve sub problems methods. These authors consider the following generating partial solutions. Merging these categories: partial solutions is iteratively applied 1) Exact algorithms to explore large within a metaheuristics. In evolutionary neighbourhoods within local search. algorithms where solutions are 2) Information of high quality solutions found in incompletely represented in the several runs of local search is used to define chromosome, exact algorithms are used as smaller problems solvable by exact algorithms. decoders to find the correspondent best 3) Exploit bounds in constructive heuristics. solution. 2.2. Incorporating metaheuristics in exact 1 algorithms. Metaheuristics can be used to Universidade do Algarve, Faro Portugal. E-mail: [email protected] obtain incumbent solutions and bounds. In 2 Univertitat Pompeu Fabra, Barcelona, Spain. E-mail: helena. [email protected] branch-and-cut and branch-and-price algorithms, metaheuristics are used to Here, we include all works of item 3 of the dynamically separate cutting-planes and classification of Dumitrescu and Stützle. pricing columns, respectively. We believe item 2.1. related with merging Metaheuristics can also be used for solutions should be generalized and renamed exactly strategic guidance of exact algorithms, or solving sub problems. to determine the branching strategy in We can say that the classification of Dumitrescu branch-and-bound techniques. and Stützle is more specific and the one of Puchinger and Raidl is more general. III. CONNECTING THE CLASSIFICATION OF DUMITRESCU AND STÜTZLE WITH THE ONE OF IV. APPLICATION TO COMBINATORIAL PUCHINGER AND RAIDL OPTIMIZATION PROBLEMS Almost all items in the classification of Dumitrescu and Stützle correspond to sub items of Using the classification of NP optimization item 2.1 (incorporating exact algorithms in problems proposed by Crescenci and Kann in metaheuristics), of the classification of Puchinger http://www.nada.kth.se/~viggo/problemlist/, we and Raidl. The exceptions are procedures classified show the distribution of the OSH heuristics by Dumitrescu and Stützle in item 2. (information of application to the different combinatorial high quality solutions found in several runs of local optimization problems. In table II we present the search is used to define smaller problems solvable problem type along with the reference of the work by exact algorithms), that have a sequential nature, where the OSH method is presented. running a local search based heuristic several times We can see that a lot of the research of before an exact algorithm; and also the work of procedures that combine metaheuristics with exact (Umetani et al. 2003), allocated to item 4 of algorithms has been dedicated to the job shop Dumitrescu et al., that sequentially executes tabu scheduling problem and to routing problems. search after solving the integer programming Packing problems and the multiple constraint relaxation. knapsack problem have also received some considerable attention, as well as the more general class of mixed integer programming problems. We TABLA I believe this can be viewed as a measurement of both CORRESPONDENCE BETWEEN THE DUMITRESCU AND the difficulty and the practical relevance of these STÜTZLE (2003) AND PUCHINGER AND RAILD (2005) problems. Practitioners are still not satisfied with the CLASSIFICATIONS results achieved by traditional applications from Classification of Dumitrescu Classification of Puchinger stand-alone fields of knowledge. and Stützle and Raidl When looking at the type of combination 1. Exact algorithms to explore 2.1 Exactly searching large implemented, we see that the most popular are large neighborhoods within neighborhoods. Merging sequential execution, exactly searching large local search solutions – exactly solving sub neighbourhoods (and here dynamic programming is problems the most used exact algorithm) and exactly solving 2. Information of high quality 1. Sequential execution solutions found in several runs subproblems. Genetic algorithms have been the of local search is used to metaheuristics procedures more frequently used in define smaller problems combination with exact algorithms, maybe because solvable by exact algorithms of it’s low performance on their one. 3. Exploit bounds in 2.1 Exact algorithms for The most common exact algorithms in this OSH constructive heuristics. strategic guidance of metaheuristics procedures are, aside from dynamic programming, 4. Local search guided by 1. Sequential execution linear relaxations and branch-and-bound. information from integer We believe using exact algorithms for strategic programming relaxations guidance of metaheuristics to be a very promising 5. Use exact algorithms for 2.1 Merging solutions specific procedures within line of research. This way we can profit from the metaheuristics fast search of the space of solutions of the metaheuristics without getting lost in a “wandering” Some works included in item 1 of the path, because of the guidance given by the exact classification of Dumitrescu et al., exactly searching algorithms. We find that another very interesting large neighbourhoods, can be viewed as a merging idea is the one of “applying the spirit of solutions kind of procedure. metaheuristics” when designing exact algorithms. We introduce a new item in the classification of Puchinger and Raidl, 2.1. related with exact algorithms for strategic guidance of metaheuristics. TABLA II CLASSIFICATION OF ARTICLES PER TYPE OF PROBLEM Problems Classified articles Problems Classified articles Mixed Integer 1.1 Sequential execution Vehicle Routing 1.1 Sequential execution (Pedroso 2004) (Ibaraki et al. 2001) 2.1.3 Exactly solving sub problems 2.1.2 Exactly searching large (Pedroso 2004) neighbourhoods 2.2.3 Metaheuristics for strategic (Thompson et al. 1989) guidance of exact algorithms (Thompson et al. 1993) (French et al. 2001) 2.1.3 Exactly solving sub problems (Kostikas et al. 2004) (Shaw 1998) 2.2.4 Applying the spirit of 2.2.4 Applying the spirit of metaheuristics metaheuristics (Danna et al. 2005) (Danna et al. 2005) (Fischetti et al. 2003) Packing 2.1.4 Exact algorithms as decoders (Puchinger et al. 2004) Graph Colouring 2.1.3 Exactly solving sub problems (Imahori