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RISS 인기검색어
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- 저자 : Xiangde Zhang (검색결과 4 건)
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On cliques and Lagrangians of hypergraphs
Qingsong Tang,Xiangde Zhang,Cheng Zhao 대한수학회 2019 대한수학회보 Vol.56 No.3
Given a graph $G$, the Motzkin and Straus formulation of the maximum clique problem is the quadratic program (QP) formed from the adjacent matrix of the graph $G$ over the standard simplex. It is well-known that the global optimum value of this QP (called Lagrangian) corresponds to the clique number of a graph. It is useful in practice if similar results hold for hypergraphs. In this paper, we attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques when the number of edges is in a certain range. Specifically, we obtain upper bounds for the Lagrangian of a hypergraph when the number of edges is in a certain range. These results further support a conjecture introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). We also establish an upper bound of the clique number in terms of Lagrangians for hypergraphs.
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ON CLIQUES AND LAGRANGIANS OF HYPERGRAPHS
Tang, Qingsong,Zhang, Xiangde,Zhao, Cheng Korean Mathematical Society 2019 대한수학회보 Vol.56 No.3
Given a graph G, the Motzkin and Straus formulation of the maximum clique problem is the quadratic program (QP) formed from the adjacent matrix of the graph G over the standard simplex. It is well-known that the global optimum value of this QP (called Lagrangian) corresponds to the clique number of a graph. It is useful in practice if similar results hold for hypergraphs. In this paper, we attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques when the number of edges is in a certain range. Specifically, we obtain upper bounds for the Lagrangian of a hypergraph when the number of edges is in a certain range. These results further support a conjecture introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). We also establish an upper bound of the clique number in terms of Lagrangians for hypergraphs.
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HOMOGENEOUS MULTILINEAR FUNCTIONS ON HYPERGRAPH CLIQUES
Lu, Xiaojun,Tang, Qingsong,Zhang, Xiangde,Zhao, Cheng Korean Mathematical Society 2017 대한수학회보 Vol.54 No.3
Motzkin and Straus established a close connection between the maximum clique problem and a solution (namely graph-Lagrangian) to the maximum value of a class of homogeneous quadratic multilinear functions over the standard simplex of the Euclidean space in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique problem in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we develop a homogeneous multilinear function based on the structure of hypergraphs and their complement hypergraphs. Its maximum value generalizes the graph-Lagrangian. Specifically, we establish a connection between the clique number and the generalized graph-Lagrangian of 3-uniform graphs, which supports the conjecture posed in this paper.
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Homogeneous multilinear functions on hypergraph cliques
Xiaojun Lu,Qingsong Tang,Xiangde Zhang,Cheng Zhao 대한수학회 2017 대한수학회보 Vol.54 No.3
Motzkin and Straus established a close connection between the maximum clique problem and a solution (namely graph-Lagrangian) to the maximum value of a class of homogeneous quadratic multilinear functions over the standard simplex of the Euclidean space in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique problem in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we develop a homogeneous multilinear function based on the structure of hypergraphs and their complement hypergraphs. Its maximum value generalizes the graph-Lagrangian. Specifically, we establish a connection between the clique number and the generalized graph-Lagrangian of $3$-uniform graphs, which supports the conjecture posed in this paper.
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