# Less Is More: Tabu Search for Bipartite Quadratic Programming Problem

## Abstract

Having defined a complete bipartite graph *G*, with weights associated with both vertices and edges, the Bipartite Quadratic Programming problem (BQP) consists in selecting a subgraph that maximizes the sum of the weights associated with the chosen vertices and the edges that connect them. Applications of the BQP arise in mining discrete patterns from binary data, approximation of matrices by rank-one binary matrices, computation of the cut-norm of a matrix, etc. In addition, BQP is also known in the literature under different names such as maximum weighted induced subgraph, maximum weight bi-clique and maximum cut on bipartite graphs. Since the problem is NP-hard, many heuristic methods have been proposed in the literature to solve it. In this paper, we apply the recent Less is more approach, whose basic idea is to design a heuristic as simple as possible, i.e., a method that uses a minimum number of ingredients but provides solutions of better quality than the current state-of-the-art. To reach that goal, we propose a simple hybrid heuristic based on Tabu search, that uses two neighborhood structures and relatively simple rule for implementation of short-term memory operation. In addition, a simple rule for calculation of tabu list length is introduced. Computational results were compared favorably with the current state-of-the-art heuristics. Despite its simplicity, our heuristic was able to find 6 new best known solutions on very well studied test instances.

## Keywords

Discrete optimization Graphs Bipartite quadratic programming Tabu search Variable neighborhood search## Notes

### Acknowledgements

The research has been supported in part by Research Grants 174010 and III 044006 of the Serbian Ministry of Education, Science and Technological Development. The research is also partially covered by the framework of the Grant Number BR05236839 “Development of information technologies and systems for stimulation of personality’s sustainable development as one of the bases of development of digital Kazakhstan”. The research is partly supported by the Ministry of Education and Science, Republic of Kazakhstan (Institute of Information and Computer Technologies), project no. AP05133090.

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