A study on the complexity of TSP instances under the 2-exchange neighbor system

This work is related to the search of complexity measures for instances of combinatorial optimization problems. Particularly, we have carried out a study about the complexity of random instances of the Traveling Salesman Problem under the 2-exchange neighbor system. We have proposed two descriptors of complexity: the proportion of the size of the basin of attraction of the global optimum over the size of the search space and the proportion of the number of different local optima over the size of the search space. We have analyzed the evolution of these descriptors as the size of the problem grows. After that, and using our complexity measures, we find a phase transition phenomenon in the complexity of the instances.

[1]  L. Kallel,et al.  How to detect all maxima of a function , 2001 .

[2]  Sébastien Vérel,et al.  Complex-network analysis of combinatorial spaces: The NK landscape case , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[4]  Philippe Collard,et al.  Fitness Distance Correlation, as statistical measure of Genetic Algorithm difficulty, revisited , 1998, ECAI.

[5]  Tad Hogg,et al.  Phase Transitions and the Search Problem , 1996, Artif. Intell..

[6]  Rich Caruana,et al.  Estimating the Number of Local Minima in Big, Nasty Search Spaces , 1999 .

[7]  Anton V. Eremeev,et al.  On Confidence Intervals for the Number of Local Optima , 2003, EvoWorkshops.

[8]  Bart Naudts,et al.  A comparison of predictive measures of problem difficulty in evolutionary algorithms , 2000, IEEE Trans. Evol. Comput..

[9]  Isaac E. Lagaris,et al.  Stopping rules for box-constrained stochastic global optimization , 2008, Appl. Math. Comput..

[10]  Rémi Monasson,et al.  Determining computational complexity from characteristic ‘phase transitions’ , 1999, Nature.

[11]  Toby Walsh,et al.  The TSP Phase Transition , 1996, Artif. Intell..

[12]  Botond Draskoczy.  Fitness Distance Correlation and Search Space Analysis for Permutation Based Problems , 2010, EvoCOP.

[13]  Assaf Naor,et al.  Rigorous location of phase transitions in hard optimization problems , 2005, Nature.

[14]  Peter F. Stadler,et al.  Towards a theory of landscapes , 1995 .

[15]  Wim Hordijk,et al.  Phase transition and landscape statistics of the number partitioning problem. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Vassilis Zissimopoulos,et al.  Autocorrelation Coefficient for the Graph Bipartitioning Problem , 1998, Theor. Comput. Sci..

[17]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[18]  Weixiong Zhang,et al.  Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem , 2011, J. Artif. Intell. Res..

[19]  Lov K. Grover.  Local search and the local structure of NP-complete problems , 1992, Oper. Res. Lett..

[20]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[21]  Alexander K. Hartmann and Martin Weigt,et al.  Phase transitions in combinatorial optimization problems , 2013 .

[22]  Andrew M. Sutton,et al.  Understanding elementary landscapes , 2008, GECCO '08.

[23]  Thomas Stützle,et al.  A review of metrics on permutations for search landscape analysis , 2007, Comput. Oper. Res..

[24]  Panos M. Pardalos,et al.  On the number of local minima for the multidimensional assignment problem , 2006, J. Comb. Optim..

[25]  Bart Selman,et al.  Computational science: Can get satisfaction , 2005, Nature.

[26]  Anton V. Eremeev,et al.  Non-parametric Estimation of Properties of Combinatorial Landscapes , 2002, EvoWorkshops.