Journal of Combinatorial Optimization

, Volume 35, Issue 4, pp 1250–1260 | Cite as

A continuous characterization of the maximum vertex-weighted clique in hypergraphs

  • Qingsong Tang
  • Xiangde Zhang
  • Guoren Wang
  • Cheng Zhao


For a simple graph G on n vertices with adjacency matrix A, Motzkin and Strauss established a remarkable connection between the clique number and the global maximum value of the quadratic programm: \(\textit{max}\{ \mathbf {x}^T A \mathbf {x}\}\) on the standard simplex: \(\{\sum _{i=1}^{n} x_i =1, x_i \ge 0 \}\). In Gibbons et al. (Math Oper Res 122:754–768, 1997), an extension of the Motzkin–Straus formulation was provided for the vertex-weighted clique number of a graph. In this paper, we provide a continuous characterization of the maximum vertex-weighted clique problem for vertex-weighted uniform hypergraphs.


Vertex-weighted hypergraphs Cliques of hypergraphs Polynomial optimization 

Mathematics Subject Classification

90C27 05C65 



We thank the anonymous referee for helpful comments. This research is partially supported by Chinese Universities Scientific Fund (No. N140504004) and the Doctoral Starting up Foundation of Liaoning Province (No. 201601011).


  1. Bomze IM (1997) Evolution towards the maximum clique. J Glob Optim 10:143–164MathSciNetCrossRefzbMATHGoogle Scholar
  2. Budinich M (2003) Exact bounds on the order of the maximum clique of a graph. Discrete Appl Math 127:535–543MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bulò SR, Pelillo M (2008) A continuous characterization of maximal cliques in \(k\)-uniform hypergraphs. In: Learning and intelligent optimization (lecture notes in computer science), vol 5315, pp 220–233Google Scholar
  4. Bulò SR, Pelillo M (2009) A generalization of the Motzkin–Straus theorem to hypergraphs. Optim Lett 3:287–295MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bulò SR, Torsello A, Pelillo M (2007) A continuous-based approach for partial clique enumeration. Graph Based Represent Pattern Recognit 4538:61–70CrossRefzbMATHGoogle Scholar
  6. Busygin S (2006) A new trust region technique for the maximum weight clique problem. Discrete Appl Math 154:2080–2096MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dong C, Zhou Q, Cai Y, Hong X (2008) Hypergraph partitioning satisfying dual constraints on vertex and edge weight. In: 51st Midwest symposium on circuits and systems, 2008. MWSCAS 2008. IEEE, pp 85–88Google Scholar
  8. Gibbons LE, Hearn DW, Pardalos PM, Ramana MV (1997) Continuous characterizations of the maximum clique problem. Math Oper Res 122:754–768MathSciNetCrossRefzbMATHGoogle Scholar
  9. Jean B, Lasserre (2001) Global optimization with polynimal and the problem of moments. SLAM J Optim 3:796–817zbMATHGoogle Scholar
  10. Klerk ED, Laurent M, Parrilo PA (2006) A PTAS for the minimization of polynomials of fixed degree over the simplex. Theor Comput Sci 361:210–225MathSciNetCrossRefzbMATHGoogle Scholar
  11. Lovasz L (1994) Stable sets and polynomials. Discrete Math 124:137–153MathSciNetCrossRefzbMATHGoogle Scholar
  12. Luenberger DG, Ye Y (2008) Linear and nonlinear programming, 3rd edn. Springer, LLC, ReadingzbMATHGoogle Scholar
  13. Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of Turán. Can J Math 17:533–540CrossRefzbMATHGoogle Scholar
  14. Pardalos PM, Phillips A (1990) A global optimization approach for solving the maximum clique problem. Int J Comput Math 33:209–216CrossRefzbMATHGoogle Scholar
  15. Pavan M, Pelillo M (2007) Dominant sets and pairwise clustering. IEEE Trans Pattern Anal Mach 29:167–172CrossRefGoogle Scholar
  16. Su L, Gao Y, Zhao X, Wan H, Gu M, Sun J (2017) Vertex-weighted hypergraph learning for multi-view object classification. In: Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI-17), pp 2779–2785Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Qingsong Tang
    • 1
  • Xiangde Zhang
    • 1
  • Guoren Wang
    • 2
  • Cheng Zhao
    • 3
  1. 1.College of SciencesNortheastern UniversityShenyangPeople’s Republic of China
  2. 2.School of Computer Science and EngineeringShenyangPeople’s Republic of China
  3. 3.Department of Mathematics and Computer ScienceIndiana State UniversityTerre HauteUSA

Personalised recommendations