Hard Problems In Box-Totally Dual Integral Polyhedra
Résumé
In this paper, we study the complexity of some fundamental questions regarding box-totally dual integral (box-TDI) polyhedra. First, although box-TDI polyhedra have strong integrality properties, we prove that Integer Programming over box-TDI polyhedra is NP-complete, that is, finding an integer point optimizing a linear function over a box-TDI polyhedron is hard. Second, we complement the result of Ding et al. [10] who proved that deciding whether a given system is box-TDI is co-NPcomplete: we prove that recognizing whether a polyhedron is box-TDI is co-NP-complete. To derive these complexity results, we exhibit new classes of totally equimodular matrices-a generalization of totally unimodular matrices-by characterizing the total equimodularity of incidence matrices of graphs.
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