Kernelization lower bound for Permutation Pattern Matching

We prove that the Permutation Pattern Matching problem does not have a polynomial kernel (assuming a widely believed hypothesis from theory of computational complexity).A novel polynomial reduction from the Clique problem to the Permutation Pattern Matching problem is introduced.The standard cross-composition framework is applied to yield polynomial lower bounds. A permutation π contains a permutation ? as a pattern if it contains a subsequence of length | ? | whose elements are in the same relative order as in the permutation ?. This notion plays a major role in enumerative combinatorics. We prove that the problem does not have a polynomial kernel (under the widely believed complexity assumption NP ? co- NP / poly ) by introducing a new polynomial reduction from the clique problem to permutation pattern matching.

[1]  Louis Ibarra.  Finding Pattern Matchings for Permutations , 1997, Inf. Process. Lett..

[2]  Donald E. Knuth,et al.  The art of computer programming: V.1.: Fundamental algorithms , 1997 .

[3]  Mike D. Atkinson,et al.  Restricted permutations , 1999, Discret. Math..

[4]  Doron Rotem,et al.  Stack sortable permutations , 1981, Discret. Math..

[5]  Michael R. Fellows,et al.  On problems without polynomial kernels , 2009, J. Comput. Syst. Sci..

[6]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[7]  Prosenjit Bose,et al.  Pattern Matching for Permutations , 1993, WADS.

[8]  Yuri Rabinovich,et al.  On Complexity of the Subpattern Problem , 2008, SIAM J. Discret. Math..

[9]  Martin Lackner,et al.  A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs , 2012, Algorithmica.

[10]  Martin Lackner,et al.  The computational landscape of permutation patterns , 2013, ArXiv.

[11]  Stefan Kratsch,et al.  Cross-Composition: A New Technique for Kernelization Lower Bounds , 2011, STACS.

[12]  Gábor Tardos,et al.  Excluded permutation matrices and the Stanley-Wilf conjecture , 2004, J. Comb. Theory, Ser. A.

[13]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[14]  Lance Fortnow,et al.  Infeasibility of instance compression and succinct PCPs for NP , 2007, J. Comput. Syst. Sci..

[15]  Dániel Marx,et al.  Finding small patterns in permutations in linear time , 2013, SODA.

[16]  C. Schensted.  Longest Increasing and Decreasing Subsequences , 1961, Canadian Journal of Mathematics.

[17]  Jacob Fox,et al.  Stanley-Wilf limits are typically exponential , 2013, ArXiv.

[18]  Martin Klazar,et al.  The Füredi-Hajnal Conjecture Implies the Stanley-Wilf Conjecture , 2000 .