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Submodular function minimization in Zn and searching in Monge arrays
Electronic Notes in Discrete Mathematics 17 (2004) 5 www.elsevier.com/locate/endm
Submodular function minimization in Zn and searching in Monge arrays Maurice Queyranne1 Sauder School of Business, University of British Columbia, Vancouver, B.C., Canada Laboratoire Leibniz-IMAG, Grenoble, France
We establish and use connections among the problem of searching in a multidimensional Monge array, minimizing a submodular function on a sublattice of the integer lattice Zn , and Boolean submodular function minimization. In particular, we obtain the first algorithm for minimizing a submodular function on a sublattice S of Zn with time complexity polynomial in the length of a maximal chain of S, and we use it to construct the first polynomial algorithm (regarded as a ”major improvement” by Aggarwal and Park) for the problem of searching in a multidimensional Monge array. We also present some results on the extension of submodular and modular functions, correcting an extension proposed by Lovasz, and we show that every modular function on a sublattice of Zn is separable. Finally, we describe a penalization approach to the problem of minimizing submodular functions on a sublattice of Zn or of the Boolean hypercube. The presentation is based on joint work with Fabio Tardella.
1
Email: [email protected]
1571-0653/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.endm.2004.03.003
Submodular function minimization in Zn and searching in Monge arrays Maurice Queyranne1 Sauder School of Business, University of British Columbia, Vancouver, B.C., Canada Laboratoire Leibniz-IMAG, Grenoble, France
We establish and use connections among the problem of searching in a multidimensional Monge array, minimizing a submodular function on a sublattice of the integer lattice Zn , and Boolean submodular function minimization. In particular, we obtain the first algorithm for minimizing a submodular function on a sublattice S of Zn with time complexity polynomial in the length of a maximal chain of S, and we use it to construct the first polynomial algorithm (regarded as a ”major improvement” by Aggarwal and Park) for the problem of searching in a multidimensional Monge array. We also present some results on the extension of submodular and modular functions, correcting an extension proposed by Lovasz, and we show that every modular function on a sublattice of Zn is separable. Finally, we describe a penalization approach to the problem of minimizing submodular functions on a sublattice of Zn or of the Boolean hypercube. The presentation is based on joint work with Fabio Tardella.
1
Email: [email protected]
1571-0653/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.endm.2004.03.003