Gouveia, João; Grappe, Roland; Kaibel, Volker; Pashkovich, Kanstantsin; Robinson, Richard Z.; Thomas, Rekha R. Which nonnegative matrices are slack matrices? (English) Zbl 1283.15103 Linear Algebra Appl. 439, No. 10, 2921-2933 (2013). Summary: We characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown. Cited in 8 Documents MSC: 15B48 Positive matrices and their generalizations; cones of matrices 52B11 \(n\)-dimensional polytopes 65F30 Other matrix algorithms (MSC2010) Keywords:slack matrices; polyhedral cones; polytopes; polyhedral verification problem; algorithm PDF BibTeX XML Cite \textit{J. Gouveia} et al., Linear Algebra Appl. 439, No. 10, 2921--2933 (2013; Zbl 1283.15103) Full Text: DOI arXiv References: [1] Björner, A., Topological methods, (Handbook of Combinatorics, (1995), Elsevier), 1819-1872 · Zbl 0851.52016 [2] Bredon, G. E., Topology and geometry, Grad. Texts in Math., vol. 139, (1993), Springer · Zbl 0791.55001 [3] Dyer, M. E., The complexity of vertex enumeration methods, Math. Oper. Res., 8, 3, 381-402, (1983) · Zbl 0531.90064 [4] Freund, R.; Orlin, J., On the complexity of four polyhedral set containment problems, Math. Program., 33, 133-145, (1985) · Zbl 0581.90060 [5] Gouveia, J.; Parrilo, P. A.; Thomas, R. R., Lifts of convex sets and cone factorizations, Math. Oper. Res., 38, 248-264, (2013) · Zbl 1291.90172 [6] Joswig, M., Software, (Goodman, Jacob E.; ORourke, Joseph, Handbook of Discrete and Computational Geometry, (2004), CRC Press), 1415-1433, Chapter 64 [7] Joswig, M.; Ziegler, G. M., Convex hulls, oracles, and homology, J. Symbolic Comput., 38, 4, 1247-1259, (2004) · Zbl 1121.52030 [8] Kaibel, V.; Pfetsch, M. E., Some algorithmic problems in polytope theory, (Algebra, Geometry, and Software Systems, (2003), Springer Berlin), 23-47 · Zbl 1027.65032 [9] Matheiss, T. H.; Rubin, D., A survey and comparison of methods for finding all vertices of convex polyhedral sets, Math. Oper. Res., 5, 2, 167-185, (1980) · Zbl 0442.90050 [10] Richter-Gebert, J., Realization spaces of polytopes, Lecture Notes in Math., vol. 1643, (1996), Springer · Zbl 0866.52009 [11] Seidel, R., Convex hull computations, (Goodman, Jacob E.; ORourke, Joseph, Handbook of Discrete and Computational Geometry, (2004), CRC Press), 495-512, Chapter 22 [12] Seymour, P. D., A note on hyperplane generation, J. Combin. Theory Ser. B, 61, 1, 88-91, (1994) · Zbl 0803.05015 [13] Steinitz, E.; Rademacher, H., Vorlesungen über die theorie der polyeder, (1934), Springer, (reprint 1976) · Zbl 0325.52001 [14] Yannakakis, M., Expressing combinatorial optimization problems by linear programs, J. Comput. System Sci., 43, 3, 441-466, (1991) · Zbl 0748.90074 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.