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A Sharper Local Lemma with Improved Applications

A Sharper Local Lemma with Improved Applications. Kashyap Kolikapa, Mario Szegedy, Yixin Xu Rutgers University APPROX&RANDOM 2012. Introduction: Lovasz Local Lemma. Independent case: . A set of bad events to avoid: . is independent of.

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A Sharper Local Lemma with Improved Applications

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  1. A Sharper Local Lemma with Improved Applications Kashyap Kolikapa, Mario Szegedy, Yixin Xu Rutgers University APPROX&RANDOM 2012

  2. Introduction: Lovasz Local Lemma Independent case: A set of bad events to avoid: is independent of General case: Lovasz Local Lemma: (symmetric version) Dependency graph

  3. Introduction: Lovasz Local Lemma Independent case: A set of bad events to avoid: is independent of General case: Dependency graph Asymmetric Version(ErdősandLovász, [EL75]).

  4. Classical Application of LLL A set of bad events to avoid: Variable Framework: Dependency graph: iff Example: Remark: Given a k-SAT instance, each clause shares variables with < other clauses then the instance is satisfiable.

  5. Cons of LLL LLL is not tight! LLL: Union bound: Is there a tight condition for ? Yes, Shearer’s condition! Roughly speaking, independent sets of dependency graph give all information!

  6. Shearer’s condition Shearer’s polynomial: Shearer’s condition: a set of events to avoid, G the dependency graph:

  7. Cons of LLL LLL is not tight! LLL: Union bound: Shearer’s condition:

  8. Motivation • Lovasz Local Lemma • Sufficient condition, not tight • Local information • Shearer’s condition • Iff condition, tight • Global information, NP-hard to compute • Is there a continuum between LLL and Shearer’s condition?

  9. Clique Lovasz Local Lemma LLL: CLLL:

  10. Comparison between LLL and CLLL • CLLL always give better bounds than LLL • LLL is a special case that cliques are single edges • Example: max degree d graph • LLL: • CLLL:

  11. Can we extend previous idea further?

  12. Reason behind LLL and CLLL LLL: CLLL: Graph Decomposition: Given an undirected graph G, a set of induced subgraphs {G1,G2, . . . ,Gm} is called a decomposition if they cover all the edges of G. Question: Can we use a different decomposition in order to get a continuum between LLL and Shearer’s condition? Answer: YES!

  13. The Decomposition Theorem a set of events to avoid, a decomposition of dependency graph G. Decompostion theorem:

  14. Applications • k-SAT • Acyclic Edge Coloring • Non-repetive Vertex Coloring • Statistical Mechacnics • Open problems: more application? • Thank you!

  15. Application to Statistical Mechanics Uniform Shearer’s Bound: s.t. satisfies Shearer’s condition. [SS06] Phase transitions in the hard-core lattice gas model [Tod99] [SS06]

  16. Application to Statistical Mechanics s.t.

  17. Application to Statistical Mechanics

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