Historical Notes on Hybrid Metaheuristics

1.1 INTRODUCTION Finding optimal solutions is in general computationally intractable for many com-binatorial optimization problems, e.g., those known as NP-hard [54]. The classical approach for dealing with this fact was the use of approximation algorithms, i.e., relaxing the goal from finding the optimal solution to obtaining solutions within some bounded distance from the former [61]. Unfortunately, it turns out that attainable bounds in practice (that is, at a tenable computational cost) are in general too far from the optimum to be useful in many problems. The days in which researchers struggled to slightly tighten worst-case bounds that were anyway far from practical, or in which finding a PTAS (let alone a FPTAS) for a certain problem was considered a whole success are thus swiftly coming to an end. Indeed, two new alternative lines of attack are being currently used to treat these difficulties. On one hand, a new corpus of theory is being built around the notion of fixed-parameter tractability that emanates from the field of parameterized complexity [40][41]. On the other hand, metaheuristics approaches are being increasingly used nowadays. Quoting [27], the philosophy of these latter techniques is " try to obtain probably optimal solutions to your problem, for provably good solutions are overwhelmingly hard to obtain ". See also [27][97] for some prospects on the intersection of both fields (parameterized complexity and metaheuristics). Focusing on the latter techniques, metaheuristics approaches can be broadly categorized into two major classes: single-solution search algorithms (also known as trajectory-based or local-search based algorithms), and multiple-solution search algorithms (also-known as population-based or –arguably stretching the term– evolutionary algorithms). Examples of the former class are descent local search (LS) [109], greedy heuristics (GH) [81], simulated annealing (SA) [72], or tabu search (TS) [56]. Among the latter class, one can cite genetic algorithms (GA) [63], evolution strategies (ES) [115], genetic programming (GP) [74], ant colony optimization (ACO) [22], scatter search (SS) [55], estimation of distribution algorithms (EDAs) [79], and others. We refer the reader to [23][58][108][116] for good overviews of metaheuristics. Over the years, interest in metaheuristics has risen considerably among researchers in combinatorial optimization. The flexibility of these techniques makes them prime candidates for tackling both new problems and variants of exiting problems. This fact,