Polyhedral Characteristics of Balanced and Unbalanced Bipartite Subgraph Problems
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We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem, we consider a balanced biclique with the same number of vertices in both parts and arbitrary edge weights. In the other two problems we are dealing with unbalanced subgraphs of maximum and minimum weight with non-negative edges. All three problems are established to be NP-hard. We study the polytopes and the cone decompositions of these problems and their 1-skeletons. We describe the adjacency criterion in the 1-skeleton of the polytope of the balanced complete bipartite subgraph problem. The clique number of the 1-skeleton is estimated from below by a superpolynomial function. For both unbalanced biclique problems we establish the superpolynomial lower bounds on the clique numbers of the graphs of nonnegative cone decompositions. These values characterize the time complexity in a broad class of algorithms based on linear comparisons.
Keywordsbiclique 1-skeleton cone decomposition clique number NP-hard problem
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- 4.Bondarenko, V.A. and Maksimenko, A.N., Geometricheskie konstruktsii i slozhnost’ v kombinatornoi optimizatsii (Geometric Constructions and Complexity in Combinatorial Optimization), Moscow: LKI, 2008.Google Scholar
- 6.Bondarenko, V. and Nikolaev, A., On graphs of the cone decompositions for the min-cut and max-cut problems, Int. J. Math. Math. Sci., 2016, vol. 2016.Google Scholar
- 8.Cheng, Y. and Church, G.M., Biclustering of expression data, Proceedings of the Eighth International Conference on Intelligent Systems for Molecular Biology, 2000, pp. 93–103Google Scholar
- 10.Feige, U. and Kogan, S., Hardness of Approximation of the Balanced Complete Bipartite Subgraph Problem. Tech. Rep. MCS04-04, The Weizmann Inst. of Science, 2004.Google Scholar
- 16.Maksimenko, A.N., Combinatorial properties of the polyhedron associated with the shortest path problem, Comput. Math. Math. Phys., 2013, vol. 88, no. 2, pp. 1611–1614.Google Scholar