A constructive algorithm for the Lovász Local Lemma on permutations

While there has been significant progress on algorithmic aspects of the Lovasz Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations: the "lopsided" version of the LLL is usually at play here, and we do not yet have subexponential-time algorithms. We resolve this by developing a randomized polynomial-time algorithm for such applications. A noteworthy application is for Latin Transversals: the best-known general result here (Bissacot et al., improving on Erdos and Spencer), states that any n x n matrix in which each entry appears at most (27/256)n times, has a Latin transversal. We present the first polynomial-time algorithm to construct such a transversal. Our approach also yields RNC algorithms: for Latin transversals, as well as the first efficient ones for the strong chromatic number and (special cases of) acyclic edge-coloring.

[1]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[2]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[3]  Pooya Hatami,et al.  A lower bound for the length of a partial transversal in a Latin square , 2008, J. Comb. Theory, Ser. A.

[4]  S. Stein.  TRANSVERSALS OF LATIN SQUARES AND THEIR GENERALIZATIONS , 1975 .

[5]  Paul Erdös,et al.  Lopsided Lovász Local Lemma and Latin transversals , 1991, Discret. Appl. Math..

[6]  Aldo Procacci,et al.  Improved bounds on coloring of graphs , 2010, Eur. J. Comb..

[7]  Noga Alon,et al.  Covering with Latin Transversals , 1995, Discret. Appl. Math..

[8]  Carsten Thomassen,et al.  Path and cycle sub-ramsey numbers and an edge-colouring conjecture , 1986, Discret. Math..

[9]  Aldo Procacci,et al.  An Improvement of the Lovász Local Lemma via Cluster Expansion , 2009, Combinatorics, Probability and Computing.

[10]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[11]  Ron Aharoni,et al.  Independent systems of representatives in weighted graphs , 2007, Comb..

[12]  Alan M. Frieze,et al.  On Rainbow Trees and Cycles , 2008, Electron. J. Comb..

[13]  Jan Vondrák,et al.  An Algorithmic Proof of the Lopsided Lovász Local Lemma ( simplified and condensed into lecture notes ) , 2015 .

[14]  Maria Axenovich,et al.  On the Strong Chromatic Number of Graphs , 2006, SIAM J. Discret. Math..

[15]  P. Erdös,et al.  Has every Latin square of order n a partial Latin transversal of size n -1? , 1988 .

[16]  Cheng Yeaw Ku,et al.  A random construction for permutation codes and the covering radius , 2006, Des. Codes Cryptogr..

[17]  Wayne Goddard,et al.  Acyclic colorings of planar graphs , 1991, Discret. Math..

[18]  Aravind Srinivasan,et al.  New Constructive Aspects of the Lovasz Local Lemma , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[19]  Wesley Pegden,et al.  An Extension of the Moser-Tardos Algorithmic Local Lemma , 2011, SIAM J. Discret. Math..

[20]  Sherman Stein,et al.  Transversals in Rectangular Arrays , 2010, Am. Math. Mon..

[21]  Aline Parreau,et al.  Acyclic edge-coloring using entropy compression , 2012, Eur. J. Comb..

[22]  Andrzej Dudek,et al.  Rainbow Hamilton Cycles in Uniform Hypergraphs , 2012, Electron. J. Comb..

[23]  Penny E. Haxell.  On the Strong Chromatic Number , 2004, Comb. Probab. Comput..

[24]  A. Scott,et al.  The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma , 2003, cond-mat/0309352.

[25]  Penny E. Haxell.  An improved bound for the strong chromatic number , 2008, J. Graph Theory.

[26]  Bruce A. Reed,et al.  Further algorithmic aspects of the local lemma , 1998, STOC '98.

[27]  Dimitris Achlioptas,et al.  Random Walks That Find Perfect Objects and the Lovasz Local Lemma , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[28]  Mario Szegedy,et al.  Moser and tardos meet Lovász , 2011, STOC.

[29]  Vladimir Kolmogorov,et al.  Commutativity in the Random Walk formulation of the Lovasz Local Lemma , 2015, ArXiv.

[30]  J. Dénes,et al.  Latin squares and their applications , 1974 .

[31]  Noga Alon,et al.  Probabilistic proofs of existence of rare events , 1989 .

[32]  C. R. Subramanian,et al.  Improved bounds on acyclic edge colouring , 2007, Discret. Math..

[33]  Linyuan Lu,et al.  Quest for Negative Dependency Graphs , 2012 .

[34]  Noga Alon,et al.  Acyclic edge colorings of graphs , 2001, J. Graph Theory.

[35]  Bruce A. Reed,et al.  Acyclic Coloring of Graphs , 1991, Random Struct. Algorithms.

[36]  Uzi Vishkin,et al.  Almost Fully-parallel Parentheses Matching , 1995, Discret. Appl. Math..

[37]  Austin Tyler Mohr.  Applications of the Lopsided Lovász Local Lemma Regarding Hypergraphs , 2013 .

[38]  Michael R. Fellows.  Transversals of Vertex Partitions in Graphs , 1990, SIAM J. Discret. Math..

[39]  James B. Shearer,et al.  On a problem of spencer , 1985, Comb..

[40]  Linyuan Lu,et al.  Using Lovász Local Lemma in the Space of Random Injections , 2007, Electron. J. Comb..

[41]  Gábor Tardos,et al.  A constructive proof of the general lovász local lemma , 2009, JACM.

[42]  Alan M. Frieze,et al.  Multicoloured Hamilton Cycles , 1995, Electron. J. Comb..

[43]  Yoshiharu Kohayakawa,et al.  Properly coloured copies and rainbow copies of large graphs with small maximum degree , 2010, Random Struct. Algorithms.