Abstract
Quadratic bottleneck assignment problems (QBAP) are obtained by replacing the addition of cost terms in the objective function of a quadratic (sum) assignment problem by taking their maximum. Since the QBAP is an \(\mathcal{NP}\)-hard problem, polynimially solvable special cases of the QBAP are of interest. In this paper we specify conditions on the cost matrices of QBAP leading to special cases which can be solved to optimality in polynomial time. In particular, the following three cases are discussed: (i) any permutation is optimal (constant QBAP), (ii) a certain specified permutation is optimal (constant permutation QBAP) and (iii) the solution can be found algorithmically by a polynomial algorithm. Moreover, the max-cone of bottleneck Monge matrices is investigated, its generating matrices are identified and it is used as a tool in proving polynomiality results.
Similar content being viewed by others
References
Burkard RE, Klinz B, Rudolf R (1996) Perspectives of Monge properties in optimization. Discrete Appl Math 3:96–161
Burkard RE, Dell’Amico M, Martello S (2009) Assignment problems. SIAM, Philadelphia
Butkovič P, Schneider H, Sergeev S (2007) Generators, extremals and bases of max cones. Linear Algebra Its Appl 421:394–406
Çela E (1998) The quadratic assignment problem. Theory and algorithms. Kluwer Academic, Dordrecht
Hardy G, Littlewood JE, Pólya G (1952) Inequalities. Cambridge University Press, Cambridge
Koopmans TC, Beckmann MJ (1957) Assignment problems and the location of economic activities. Econometrica 25:53–76
Krushevski AV (1964) The linear programming problem on a permutation group. In: Proceedings of the seminar on methods of mathematical modeling and theory of electrical circuits, vol 3. Institute of Cybernetics of the Academy of Sciences of Ukraine, pp 364–371 (in Russian)
Rubinstein MI (1994) The two and four points conditions in the quadratic assignment problems. Manuscript, Moscow
Rubinstein MI (1976) Problems and methods in combinatorial programming. Technical Report, Institute of Control Sciences, Moscow (in Russian)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Burkard, R.E., Rissner, R. Polynomially solvable special cases of the quadratic bottleneck assignment problem. J Comb Optim 22, 845–856 (2011). https://doi.org/10.1007/s10878-010-9333-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-010-9333-7