Random Walks That Find Perfect Objects and the Lovasz Local Lemma

We give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly. Our work is inspired by Moser's entropic method proof of the Lovasz Local Lemma (LLL) for satisfiability and completely bypasses the Probabilistic Method formulation of the LLL. In particular, our method works when the underlying state space is entirely unstructured. Similarly to Moser's argument, the key point is that algorithmic progress is measured in terms of entropy rather than energy (number of violated constraints) so that termination can be established even under the proliferation of states in which every step of the algorithm (random walk) increases the total number of violated constraints.

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

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

[3]  Joel H. Spencer,et al.  Asymptotic lower bounds for Ramsey functions , 1977, Discret. Math..

[4]  Wesley Pegden,et al.  Highly nonrepetitive sequences: Winning strategies from the local lemma , 2010, Random Struct. Algorithms.

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

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

[7]  Mario Szegedy.  The Lovász Local Lemma - A Survey , 2013, CSR.

[8]  Dimitris Achlioptas,et al.  The Lovász Local Lemma as a Random Walk , 2014, ArXiv.

[9]  Jan Vondrák,et al.  An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

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

[11]  Christian Scheideler,et al.  Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lovász local lemma , 2000, SODA '00.

[12]  Robin A. Moser.  A constructive proof of the Lovász local lemma , 2008, STOC '09.

[13]  P. Erdos-L Lovász.  Problems and Results on 3-chromatic Hypergraphs and Some Related Questions , 2022 .

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

[15]  Artur Czumaj,et al.  Coloring nonuniform hypergraphs: a new algorithmic approach to the general Lovász local lemma , 2000 .

[16]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[17]  Noga Alon,et al.  A Parallel Algorithmic Version of the Local Lemma , 1991, Random Struct. Algorithms.

[18]  Jakub Przybylo,et al.  Can Colour-Blind Distinguish Colour Palettes? , 2013, Electron. J. Comb..

[19]  Andrzej Dudek,et al.  Extensions of Results on Rainbow Hamilton Cycles in Uniform Hypergraphs , 2015, Graphs Comb..

[20]  C.H. Papadimitriou,et al.  On selecting a satisfying truth assignment , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[21]  Karthekeyan Chandrasekaran,et al.  Deterministic algorithms for the Lovász Local Lemma , 2009, SODA '10.

[22]  Dan Suciu,et al.  Journal of the ACM , 2006 .

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

[24]  Aravind Srinivasan.  Improved algorithmic versions of the Lovász Local Lemma , 2008, SODA '08.

[25]  Bruce A. Reed,et al.  Vertex colouring edge partitions , 2005, J. Comb. Theory B.

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

[27]  Mario Szegedy,et al.  A Sharper Local Lemma with Improved Applications , 2012, APPROX-RANDOM.

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

[29]  P. Diaconis,et al.  The cutoff phenomenon in finite Markov chains. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

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

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

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