Complexity indices for the travelling salesman problem and data mining

Document Type: Research Paper

Author

Abstract

 we extend our previous work on complexity indices for the travelling salesman problem (TSP), summarized in cite{CvCK3}, using graph spectral techniques of data mining. A complexity index is an invariant of an instance $I$ by which we can predict the execution time of an exact algorithm for TSP for $I$. We consider the symmetric travelling salesman problem with instances $I$ represented by complete graphs $G$ with distances between vertices (cities) as edge weights (lengths). Intuitively, the hardness of an instance $G$ depends on the distribution of short edges within $G$. Therefore we consider some short edge subgraphs of $G$ (minimal spanning tree, critical connected subgraph, and several others) as non-weighted graphs and several their invariants as potential complexity indices. Here spectral invariants (e.g. spectral radius of the adjacency matrix) play an important role since, in general, there are intimate relations between eigenvalues and the structure of a graph. Since hidden details of short edge subgraphs really determine the hardness of the instance, we use techniques of data mining to find them. In particular, spectral clustering algorithms are used including information obtained from the spectral gap in Laplacian spectra of short edge subgraphs.

Keywords

Main Subjects

References

B. Ashok and T.K. Patra (2010). Locating phase transitions in computationally hard problems. J. Physics, to appear.
D. Cvetkovi' c, M. v Cangalovi' c and V. Kovav cevi' c-Vujv ci' c (1999). Semidefinite programming methods for the symmetric travelling salesman problem. Integer Programming and Combinatorial Optimization, Proc. 7th Internat. IPCO Conf.Graz, Austria, June 1999, Lecture Notes Comp. Sci. 1610, Springer, Berlin. , 126-136
D. Cvetkovi' c, M. v Cangalovi' c and V. Kovav cevi' c-Vujv ci' c (1999). Complexity indices for the travelling salesman problem based on a semidefinite relaxation. SYM-OP-IS '99, Proc. XXVI Yugoslav Symp. Operations Research, Beograd. , 177-180
D. Cvetkovi' c, M. v Cangalovi' c and V. Kovav cevi' c-Vujv ci' c (2004). Optimization and highly informative graph invariants. Two Topics in Mathematics, ed. B.Stankovi' c, Zbornik radova 10(18), Matemativ cki institut SANU, Beograd. , 5-39
D. Cvetkovi'c, P. Rowlinson and S. K. Simi'c (2009). An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge.
D. Cvetkovi' c and S.K. Simi' c (2011). Graph spectra in computer science. Linear Algebra Appl.. 434, 1545-1562
M. Fiedler (1973). Algebraic connectivity of graphs. Czech. J. Math.. 23 (98), 298-305
G. Gutin and A. Punnen (2002). The Travelling Salesman Problem and Its Variations. Kluwer Academic Publishers, Dordrecht.
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnoy Kan and D.B. Shmoys (1985). The Traveling Salesman Problem. John Wiley and Sons, Chichester - New York - Brisbane - Toronto - Singapore.
K. Ko, P. Orponen, U. Sch" oning and O. Watanabe (1986). What is a hard instance of a computational problem ?. Structure in Complexity Theory. Conf. 1986, Lecture Notes in Computer Science, Springer. , 197-217
U. von Luxburg (2007). A tutorial on spectral clustering. Stat. Comput.. 17, 395-416
C.R. Reeves and A.V. Eremeev (2004). Statistical analysis of local search landscapes. J. Oper. Res. Soc.. 55, 687-693
B.D. Reyck and W. Herroelen (1996). On the use of the complexity index as a measure of complexity in activity networks. Europ. J. Oper. Res.. 91, 347-366
R. Sawilla (2008). A survey of data mining of graphs using spectral graph theory. Defence R&D Canada, Ottawa, Technical Memorandum TM 2008-317, Ottawa.
Transactions on Combinatorics

Volume 1, Issue 1
March 2012
Pages 35-43

Files

Share

How to cite

Statistics