A constructive algorithm for the LLL on permutations

While there has been significant progress on algorithmic aspects of the Lov\'{a}sz Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. 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 Erd\H{o}s and Spencer), states that any $n \times n$ 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. We also develop RNC algorithms for Latin transversals, rainbow Hamiltonian cycles, strong chromatic number, and hypergraph packing. In addition to efficiently finding a configuration which avoids bad-events, the algorithm of Moser & Tardos has many powerful extensions and properties. These include a well-characterized distribution on the output distribution, parallel algorithms, and a partial resampling variant. We show that our algorithm has nearly all of the same useful properties as the Moser-Tardos algorithm, and present a comparison of this aspect with recent works on the LLL in general probability spaces.

[1]  Penny Haxell.  An improved bound for the strong chromatic number , 2008 .

[2]  Wesley Pegden.  An Extension of the Moser-Tardos Algorithmic Local Lemma , 2014, SIAM J. Discret. Math..

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

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

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

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

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

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

[9]  Aravind Srinivasan,et al.  A constructive algorithm for the Lovász Local Lemma on permutations , 2014, SODA.

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

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

[12]  Noga Alon.  The String Chromatic Number of a Graph , 1992, Random Struct. Algorithms.

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

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

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

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

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

[18]  Michael Luby.  A Simple Parallel Algorithm for the Maximal Independent Set Problem , 1986, SIAM J. Comput..

[19]  Mario Szegedy,et al.  Moser and tardos meet Lovász , 2011, STOC '11.

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

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

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

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

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

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

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

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

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

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

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

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

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