Estimating clique size via discarding subgraphs

Authors

  • Sandor Szabo Institute of Mathematics and Informatics, University of Pecs
  • Bogdan Zavalnij Alfred Renyi Institute of Mathematics

DOI:

https://doi.org/10.31449/inf.v45i2.3107

Abstract

The paper will present a method to establish an upper bound on the clique number of agiven finite graph. In order to evaluate the proposed algorithm in practice we carry outa large scale numerical experiment.

References

E. Balas and J. Xue,

emph{Weighted and unweighted maximum clique algorithms with

upper bounds from fractional coloring,}

Algorithmica 15 (1996), pp.~397--412.

I. M. Bomze, M. Budinich, P. M. Pardalos and

M. Pelillo, emph{The Maximum Clique Problem, Handbook of

Combinatorial Optimization Vol. 4,} Kluwer Academic Publisher,

B. Borchers, emph{CSDP, A C Library for Semidefinite

Programming,} Optimization Methods and Software,

(1) (1999), pp.~613--623.

B. Borchers and J. G. Young. emph{Implementation of a primal-dual method

for SDP on a shared memory parallel architecture,} Computational

Optimization and Applications 37(3), (2007) pp.~355--369.

A. B'ota and M. Kr'esz. emph{A High Resolution Clique-based

Overlapping Community Detection Algorithm for Small-world Networks.}

Informatica, 39(2). (2015).

D. Brelaz,

emph{New methods to color the vertices of a graph,}

Communications of the ACM, 22 (1979), pp.~251--256.

J. C. Culberson, emph{Iterated Greedy Graph Coloring and the

Difficulty Landscape,} Technical Report, University of

Alberta, 1992.

C. Elphick and P. Wocjan,

emph{An inertial lower bound for the chromatic number of a graph,}

arXiv:1605.01978 (2016) url{https://arxiv.org/abs/1605.01978}

M. R. Garey and D. S. Johnson,

emph{Computers and Intractability: A Guide to the Theory of NP-completeness,}

Freeman, New York, 2003.

F. Harary, emph{Graph Theory,} Addison-Wesley, 1969.

J. Hasselberg, P. M. Pardalos, and G. Vairaktarakis,

emph{Test case generators and computational results for the maximum clique

problem,}

Journal of Global Optimization 3 (1993), pp.~463--482.

url{http://www.springerlink.com/content/p2m65n57u657605n}

D. Kumlander,

emph{Some Practical Algorithms to Solve the Maximal Clique Problem,}

PhD. Thesis, Tallin University of Technology, 2005.

S. Lamm, P. Sanders, C. Schulz, D. Strash, R.

F. Werneck. Finding Near-Optimial Independent Sets at

Scale. {it Proceedings of the 16th Meeting on Algorithm Engineering and

Experimentation (ALENEX'16).} 2016.

F. T. Leighton,

emph{A graph coloring algorithm for large scheduling problems,}

Journal of Research of National Bureau of Standards

(1979), pp.~489--506.

L. Lovasz, emph{On the Shannon Capacity of a Graph,}

IEEE Transactions on Information Theory, Volume 25 Issue 1,

January 1979 pp.~1--7.

P.R.J. "Ostergaa rd and A. P"oll"anen, emph{New Results on Tripod

Packings.} Discrete Comput Geom 61, 271--284 (2019).

C. H. Papadimitriou,

emph{Computational Complexity,}

Addison-Wesley Publishing Company, Inc., Reading, MA 1994.

N. J. A. Sloane,

emph{Challenge Problems: Independent Sets in Graphs,}

url{http://neilsloane.com/doc/graphs.html}

S. Stahl, emph{$n$-Tuple colorings and associated

graphs.} Journal of Combinatorial Theory, Series B Volume 20, Issue

, April 1976, pp.~185--203.

S. Szab'o

emph{Monotonic matrices and clique search in graphs,}

Annales Univ. Sci. Budapest., Sect. Comp.

(2013), pp.~307--322.

S. Szab'o and B. Zav'alnij, emph{Reducing graph

coloring to clique search,} Asia Pacific Journal of Mathematics, 1

(2016), pp.~64--85.

S. Szab'o and B. Zav'alnij, A different

approach to maximum clique search. emph{20th International Symposium on

Symbolic and Numeric Algorithms for Scientific Computing

(SYNASC2018)} pp.~310--316.

S. Szab'o and B. Zav'alnij, emph{Reducing

hypergraph coloring to clique search.} Discrete Applied Mathematics,

(2019), pp.~196--207.

Q. Wu and J-K. Hao, emph{A review on algorithms for maximum

clique problems,} European Journal of Operational Research.

Volume 242, Issue 3, 1 May 2015, pp.~693--709.

Downloads

Published

2021-06-15

How to Cite

Szabo, S., & Zavalnij, B. (2021). Estimating clique size via discarding subgraphs. Informatica, 45(2). https://doi.org/10.31449/inf.v45i2.3107

Issue

Section

MATCOS-19