Abstract
We study the Bipartite Boolean Quadratic Programming Problem (BBQP) which is an extension of the well known Boolean Quadratic Programming Problem (BQP). Applications of the BBQP include mining discrete patterns from binary data, approximating matrices by rank-one binary matrices, computing the cut-norm of a matrix, and solving optimisation problems such as maximum weight biclique, bipartite maximum weight cut, maximum weight induced sub-graph of a bipartite graph, etc. For the BBQP, we first present several algorithmic components, specifically, hill climbers and mutations, and then show how to combine them in a high-performance metaheuristic. Instead of hand-tuning a standard metaheuristic to test the efficiency of the hybrid of the components, we chose to use an automated generation of a multi-component metaheuristic to save human time, and also improve objectivity in the analysis and comparisons of components. For this we designed a new metaheuristic schema which we call Conditional Markov Chain Search (CMCS). We show that CMCS is flexible enough to model several standard metaheuristics; this flexibility is controlled by multiple numeric parameters, and so is convenient for automated generation. We study the configurations revealed by our approach and show that the best of them outperforms the previous state-of-the-art BBQP algorithm by several orders of magnitude. In our experiments we use benchmark instances introduced in the preliminary version of this paper and described here, which have already become the de facto standard in the BBQP literature.
Highlights
The (Unconstrained) Boolean Quadratic Programming Problem (BQP) is to maximise f (x) = xT Q x + c x + c0 subject to x ∈ {0, 1}n, where Q is an n × n real matrix, c is a row vector in Rn, and c0 is a constant
We show that a special case of Conditional Markov Chain Search (CMCS) that we proposed significantly outperforms several standard metaheuristics, on this problem
Only a modest computational power was required to obtain it. (Note that this computational power should not be compared to the running time of the algorithm itself; it is a replacement of expensive time of a human expert working on manual design of a high-performance solution method.) We believe that these results strongly support the idea of automated metaheuristic in general and CMCS schema in particular
Summary
The Maximum Cut Problem on a bipartite graph (MaxCut) can be formulated as BBQP (Punnen et al, 2015b) and this gives yet another application of the model. Our main goals are to verify that the proposed components are sufficient to build a high-performance heuristic for BBQP and investigate the most promising combinations By this computational study, we further support the ideas in the areas of automated parameter tuning and algorithm configuration A set of benchmark instances is developed These test instances were first introduced in the preliminary version of this paper (Karapetyan & Punnen, 2012) and since used in a number of papers (Duarte et al, 2014; Glover et al, 2015) becoming de facto standard testbed for the BBQP.
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