Solving Large Neighborhoods or Subproblems

The general idea ofvery large(-scale) neighborhood search(VLNS) [5] is to apply a more sophisticated procedure than naive enumeration to search for a best (or better) solution within a larger but restricted part of the whole search space induced by an incumbent solution. Var-ious techniques especially including (mixed) integer programming methods, dynamic pro-gramming (e.g.Dynasearch[34]), and constraint programming have been successfully used in VLNS as embedded optimization procedures. Hu et al. [109] fully explore some neighbor-hoods of a VND via dynamic programming and ILP techniques for the generalized minimum spanning tree problem. In this vein in [174] Prandtstetter and Raidl several different MIP-based neighborhoods are searched within a VNS framework for a car sequencing problem.

Similar ideas, also utilizing ILP techniques, are applied in Section 4.9 and 6.4, though sev-eral of our approaches are somewhat a mixture of large neighborhood search and optimal solution merging.

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As already pointed out in Section 2.3.1, in cut and column generation based integer program-ming methods the dynamic separation of cutting planes and the pricing of columns can be done by means of (meta-)heuristics in order to speed up the optimization process, especially when facing difficult subproblems. An example is the branch-and-cut algorithm by Gruber and Raidl [101] for the bounded diameter minimum spanning tree problem. The diameter bound is ensured via an exponentially large number of so-called jump inequalities. A se-quence of methods is used for their separation, starting from a greedy construction technique over a local search procedure to a tabu search algorithm. In Section 4.8 we apply a heuris-tic multi-start method, basically similar to a VND, to find violated 2-path cuts for a routing problem in a B&C&P approach.

Puchinger and Raidl [184] describe an exact branch-and-price algorithm for the three-stage two-dimensional bin packing problem. The pricing problem occurring in this application is a three-stage two-dimensional knapsack packing problem. Fast column generation is per-formed by applying the following sequence: a greedy heuristic, an evolutionary algorithm, solving a restricted, simpler IP-model of the pricing problem using a MIP solver within a cer-tain time-limit, and finally solving a complete IP-model by the MIP solver. The algorithms coming later in this sequence are only executed if the previous ones did not find columns with negative reduced costs. In Section 4.7.2 we also apply several methods of increasing computational effort to solve anN P-hard pricing problem. Sometimes also CP is applied to generate colums [102].

Speaking of which, in CP a subproblem is to filter the inconsistent values of variable do-mains. Richter et al. [195] propose theSomeDifferentconstraint and heuristic filtering algorithms, whereas in [85] Galinier et al. present a tabu search to handle this filtering.

Finally we note that sometimes a naturally occurring subproblem might be solved by means of an exact technique to improve the performance of the main heuristic approach. For exam-ple Fuellerer et al. [84] solve a two-dimensional loading problem via B&B for a combined vehicle routing and loading problem. Similar to generating columns as before, the exact method is only applied in case some heuristics failed to find a solution. We apply a simi-lar scheme for a routing problem involving several compartments to be filled in Chapter 7, whereas we finally rely on CP.

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3

C ONSENSUS T REE P ROBLEM

3.1 Introduction

The consensus tree problem(CTP) has been first motivated and described in [3] alongside with a solution method. Although it most frequently appears in the domain of phyloge-netics [107], it has potential applications in other clustering domains as well. But to start with, the reason for dealing with this interesting problem was the joint project entitled Hy-bridizing Branch-and-Bound with Metaheuristics for Solving Tree-Structured Combinatorial Optimization Problemssupported by the Austrian exchange service (ÖAD) within the WTZ program (scientific and technological cooperation), where cooperations with partners from Spain were facilitated in the call “Acciones Integradas 2006–2007”. Our partners were Car-los Cotta, Antonio José Fernández and Jose Enrique Gallardo from the University of Malaga.

Among others one of the main aspects of the project was to develop new methods to espe-cially deal with the inference of phylogenetic trees. For this we agreed on a two phase approach: our Spanish colleagues aimed at deriving high quality trees, and our task was to consolidate these trees into one final output tree, hence solving the CTP.

In Section 3.2 we report on previous as well as related work. The tree similarity measures utilized in our work are defined in Section 3.3. Meaningful tree neighborhood structures for local search based approaches are presented in Section 3.4. They are applied in a variable neighborhood search (VNS) with an embedded variable neighborhood descent (VND) as described in Section 3.7. The neighborhoods are further utilized in Section 3.6 to extend an existing evolutionary algorithm (EA) by local search to a memetic algorithm (MA). Finally, in Section 3.8 we consider sequential and intertwined combinations of the EA (MA) and VNS (VND). Experimental results on real and artificially generated CTP instances are given in Section 3.11, followed by concluding remarks in Section 3.12.

3. CONSENSUSTREEPROBLEM

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root node

inner nodes leaf nodes

branches

Figure 3.1:Exemplary rooted and unrooted evolutionary trees.

Parts of this work were presented at the Austrian Workshop on Metaheuristics 5in 2007 (AWM 5 ’07), at the2nd International Conference on Bioinformatics Research and Develop-mentin 2008 (BIRD’08) [169] (also see our poster in the appendix in Section A.1), and at the 10th Annual Conference on Genetic and Evolutionary Computationin 2008 (GECCO’08) [157].