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The bilinear assignment problem: complexity and polynomially solvable special cases

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Abstract

In this paper we study the bilinear assignment problem (BAP) with size parameters m and n, \(m\le n\). BAP is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence NP-hard. We show that BAP cannot be approximated within a constant factor unless P = NP even if the associated quadratic cost matrix Q is diagonal. Further, we show that BAP remains NP-hard if \(m = O(\root r \of {n})\), for some fixed r, but is solvable in polynomial time if \(m = O\left( \frac{\log n}{\log \log n}\right) \). When the rank of Q is fixed, BAP is observed to admit FPTAS and when this rank is one, it is solvable in polynomial time under some additional restrictions. We then provide a necessary and sufficient condition for BAP to be equivalent to two linear assignment problems. A closed form expression to compute the average of the objective function values of all solutions is presented, whereas the median of the solution values cannot be identified in polynomial time, unless P = NP. We then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of \((m-1)!(n-1)!\) solutions. However, computing a solution whose objective function value is no worse than that of \(m!n!-\lceil \frac{m}{\beta }\rceil !\lceil \frac{n}{\beta }\rceil !\) solutions is NP-hard for any fixed rational number \(\beta >1\).

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Acknowledgements

This research work was supported by an NSERC discovery grant and an NSERC discovery accelerator supplement awarded to Abraham P. Punnen and an NSERC discovery grant awarded to Binay Bhattacharya. We are thankful to the referees for their comments which improved the presentation of the paper. Further, we would like to thank Gerhard J. Woeginger for his suggestions, and Stefan Lendl for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University Surrey, 250-13450 102nd AV, Surrey, British Columbia, V3T 0A3, Canada

    Ante Ćustić & Abraham P. Punnen

  2. School of Computer Science, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada

    Vladyslav Sokol & Binay Bhattacharya

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  1. Ante Ćustić
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  2. Vladyslav Sokol
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  3. Abraham P. Punnen
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Correspondence to Ante Ćustić.

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Ćustić, A., Sokol, V., Punnen, A.P. et al. The bilinear assignment problem: complexity and polynomially solvable special cases. Math. Program. 166, 185–205 (2017). https://doi.org/10.1007/s10107-017-1111-1

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