A generalization of the Motzkin–Straus theorem to hypergraphs

In 1965, Motzkin and Straus established a remarkable connection between the global maxima of the Lagrangian of a graph G over the standard simplex and the clique number of G. In this paper, we provide a generalization of the Motzkin–Straus theorem to k-uniform hypergraphs (k-graphs). Specifically, given a k-graph G, we exhibit a family of (parameterized) homogeneous polynomials whose local (global) minimizers are shown to be in one-to-one correspondence with maximal (maximum) cliques of G.

[1]  Pablo A. Parrilo,et al.  A PTAS for the minimization of polynomials of fixed degree over the simplex , 2006, Theor. Comput. Sci..

[2]  H. Wilf.  The Eigenvalues of a Graph and Its Chromatic Number , 1967 .

[3]  Marcello Pelillo,et al.  Dominant Sets and Pairwise Clustering , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  Vojtech Rödl,et al.  Hypergraphs do not jump , 1984, Comb..

[5]  Dhruv Mubayi,et al.  A hypergraph extension of Turán's theorem , 2006, J. Comb. Theory, Ser. B.

[6]  Marco Budinich.  Exact Bounds on the Order of the Maximum Clique of a Graph , 2003, Discret. Appl. Math..

[7]  Panos M. Pardalos,et al.  A continuous based heuristic for the maximum clique problem , 1993, Cliques, Coloring, and Satisfiability.

[8]  L. Baum,et al.  An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology , 1967 .

[9]  Etienne de Klerk,et al.  The complexity of optimizing over a simplex, hypercube or sphere: a short survey , 2008, Central Eur. J. Oper. Res..

[10]  M. Pelillo,et al.  Payoff-Monotonic Game Dynamics and the Maximum Clique Problem , 2006 .

[11]  Herbert S. Wilf,et al.  Spectral bounds for the clique and independence numbers of graphs , 1986, J. Comb. Theory, Ser. B.

[12]  Stanislav Busygin,et al.  A new trust region technique for the maximum weight clique problem , 2006, Discret. Appl. Math..

[13]  Andrea Torsello,et al.  Payoff-Monotonic Game Dynamics and the Maximum Clique Problem , 2006, Neural Computation.

[14]  Andrea Torsello,et al.  A Continuous-Based Approach for Partial Clique Enumeration , 2007, GbRPR.

[15]  M. Pelillo.  Relaxation labeling networks for the maximum clique problem , 1996 .

[16]  P. Pardalos,et al.  A global optimization approach for solving the maximum clique problem , 1990 .

[17]  Immanuel M. Bomze,et al.  Evolution towards the Maximum Clique , 1997, J. Glob. Optim..

[18]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[19]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[20]  A. Jagota,et al.  Feasible and infeasible maxima in a quadratic program for maximum clique , 1996 .