Abstract
We introduce a concept of dual consistency of systems of linear inequalities with full generality. We show that a cardinality constrained polytope is represented by a certain system of linear inequalities if and only if the systems of linear inequalities associated with the cardinalities are dual consistent. Typical dual consistent systems of inequalities are those which describe polymatroids, generalized polymatroids, and dual greedy polyhedra with certain choice functions. We show that the systems of inequalities for cardinality-constrained ordinary bipartite matching polytopes are not dual consistent in general, and give additional inequalities to make them dual consistent. Moreover, we show that ordinary systems of inequalities for the cardinality-constrained (poly)matroid intersection are not dual consistent, which disproves a conjecture of Maurras, Spiegelberg, and Stephan about a linear representation of the cardinality-constrained polymatroid intersection.
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Notes
A cardinality constraint in an ordinary sense has the normal vector of all ones, but we consider any non-zero vector \(z_0\) in general.
Note that \(P^{c_1}_{f_1}\) instead of \(P^{c_2}_{f_2}\) works as well.
Note that for three sets \(A\), \(B\), and \(C\) with \(A\cap B\ne \emptyset \) and \(B\cap C\ne \emptyset \) we may have \(A\cap C=\emptyset \), which explains the possible non-transitivity. See bipartite matching polytopes discussed in Sect. 4.2 as such an example.
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Acknowledgments
The first author’s research was supported by JSPS Grant-in-Aid for Scientific Research (B) 25280004. The second author’s research was done when he was working at the Research Institute for Discrete Mathematics, University of Bonn. The present research was partly supported due to the “General Agreement for Cooperation between Hausdorff Center for Mathematics, University of Bonn and Research Institute for Mathematical Sciences, Kyoto University”. The authors thank the two anonymous referees for their useful comments that improved the presentation of this paper.
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Fujishige, S., Maßberg, J. Dual consistent systems of linear inequalities and cardinality constrained polytopes. Math. Program. 150, 35–48 (2015). https://doi.org/10.1007/s10107-014-0748-2
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DOI: https://doi.org/10.1007/s10107-014-0748-2