The Lopsided Lovász Local Lemma

Transcription

1 Joint work with Linyuan Lu and László Székely Georgia Southern University April 27, 2013 The lopsided Lovász local lemma can establish the existence of objects satisfying several weakly correlated conditions simultaneously.

2 Lovász Local Lemma Given a collection of events in which most are independent of one another, there is a nonzero chance that none of them occurs.

3 2-Coloring Hypergraphs Hypergraph Ground set V Collection E of nonempty subsets of V V = {v i i [5]} E = {{v 1, v 2, v 3 }, {v 2, v 3 }, {v 3, v 4, v 5 }}

4 2-Coloring Hypergraphs A 2-coloring is an assignment of two colors to the vertices and is proper if no edge is monochromatic.

5 2-Coloring Hypergraphs If there are many intersections among small edges, a hypergraph may be impossible to 2-color properly. How many intersections can we allow and still guarantee a proper 2-coloring exists?

6 2-Coloring Hypergraphs Color the vertices of a hypergraph H with two colors independently at random. For each edge e, define the bad event A e = {2-colorings of H e is monochromatic}. The proper 2-colorability of H is equivalent to Pr > 0. e E(H) A e

7 2-Coloring Hypergraphs Let e be an edge of the hypergraph H and F be a collection of edges that are disjoint from e. The event A e is independent of the event algebra generated by {A f f F }. We capture this information in a dependency graph.

8 2-Coloring Hypergraphs Dependency Graph G [Erdős, Lovász 1975] Each vertex corresponds to an event. Each event is independent of the event algebra generated by its non-neighbors in G. For hypergraph 2-coloring, the graph G with V (G) = E(H) and is a dependency graph. E(G) = {ef e and f share a vertex in H}

9 2-Coloring Hypergraphs Graph to be 2-colored Dependency Graph

10 2-Coloring Hypergraphs Lemma (Symmetric Local Lemma - Erdős, Lovász 1975) Let {A i i [n]} be a collection of events having a dependency graph of maximum degree d. If then Pr (A i ) p for all i and ep(d + 1) 1, ( n ) Pr A i > 0. i=1

11 2-Coloring Hypergraphs Theorem (Erdős, Lovász 1975) Let H be a hypergraph in which every edge contains at least k vertices. If each edge intersects at most 2k 1 e 1 other edges, then H is properly 2-colorable.

12 2-Coloring Hypergraphs Proof. Set p = 1 and d = 2k 1 2 k 1 e 1. Dependency graph has maximum degree at most d (since each edge intersects at most 2k 1 e 1 other edges) Pr (A i ) 1 for all i (since every edge has size at least k) 2 k 1 1 ep(d + 1) = e 2k 1 2 k 1 e = 1 The symmetric local lemma gives Pr ( n i=1 A i) > 0. That is, H is properly 2-colorable.

13 2-Coloring Hypergraphs Proof. Set p = 1 and d = 2k 1 2 k 1 e 1. Dependency graph has maximum degree at most d (since each edge intersects at most 2k 1 e 1 other edges) Pr (A i ) 1 for all i (since every edge has size at least k) 2 k 1 1 ep(d + 1) = e 2k 1 2 k 1 e = 1 The symmetric local lemma gives Pr ( n i=1 A i) > 0. That is, H is properly 2-colorable.

14 2-Coloring Hypergraphs Proof. Set p = 1 and d = 2k 1 2 k 1 e 1. Dependency graph has maximum degree at most d (since each edge intersects at most 2k 1 e 1 other edges) Pr (A i ) 1 for all i (since every edge has size at least k) 2 k 1 1 ep(d + 1) = e 2k 1 2 k 1 e = 1 The symmetric local lemma gives Pr ( n i=1 A i) > 0. That is, H is properly 2-colorable.

15 2-Coloring Hypergraphs Proof. Set p = 1 and d = 2k 1 2 k 1 e 1. Dependency graph has maximum degree at most d (since each edge intersects at most 2k 1 e 1 other edges) Pr (A i ) 1 for all i (since every edge has size at least k) 2 k 1 1 ep(d + 1) = e 2k 1 2 k 1 e = 1 The symmetric local lemma gives Pr ( n i=1 A i) > 0. That is, H is properly 2-colorable.

16 2-Coloring Hypergraphs Proof. Set p = 1 and d = 2k 1 2 k 1 e 1. Dependency graph has maximum degree at most d (since each edge intersects at most 2k 1 e 1 other edges) Pr (A i ) 1 for all i (since every edge has size at least k) 2 k 1 1 ep(d + 1) = e 2k 1 2 k 1 e = 1 The symmetric local lemma gives Pr ( n i=1 A i) > 0. That is, H is properly 2-colorable.

17 Lovász Local Lemma Lemma (Asymmetric Local Lemma - Spencer 1977) Let {A i i [n]} be a collection of events having a dependency graph G. If there are real numbers x i [0, 1) such that, for all i, Pr (A i ) x i ij E(G) (1 x j ), then ( n ) n Pr A i (1 x i ) > 0. i=1 i=1 The symmetric version follows from the asymmetric version by setting each x i = 1 d+1.

18 Derangements A derangement is a permutation of [n] having no fixed point. Define the canonical event for each i [n]. A i = {permutations π of [n] π(i) = i}. The set n i=1 A i contains precisely the derangements of [n].

19 Derangements The local lemma does not apply, since no pair of the events are independent: while Pr (A 1 A 2 ) = Pr (A 1 ) Pr (A 2 ) = (n 2)! n! (n 1)! n! = 1 n 2 n, (n 1)! n! = 1 n 2.

20 Derangements Fortunately, derangements possess a different useful property: and Pr (A 1 ) = (n 1)! n! = 1 n Pr ( A 1 A 2 ) = A 1 A 2 A 2 so = (n 1)! (n 2)! n! (n 1)! Pr ( A 1 A 2 ) Pr (A1 ) = 1 n 1 n(n 1) 2 We capture this information in a negative dependency graph.

21 Derangements Negative Dependency Graph G [Erdős, Spencer 1991] Each vertex corresponds to an event. For each event A i and any subset S of its non-neighbors in G, Pr A i Pr (A i ). j S A j Theorem (Lu, Székely 2006) The graph with vertex set [n] and no edges is a negative dependency graph for the canonical events {A i i [n]}.

22 Lopsided Local Lemma Given a collection of events in which most are negative dependent, there is a nonzero chance that none of them occurs. Lemma (Lopsided Local Lemma - Erdős, Spencer 1991) Let {A i i [n]} be a collection of events having negative dependency graph G. If there are real numbers x i [0, 1) such that, for all i, Pr (A i ) x i ij E(G) (1 x j ), then ( n ) n Pr A i (1 x i ) > 0. i=1 i=1

23 Derangements For derangements, setting each x i = 1 n in the lopsided local lemma gives ( n ) ( Pr A i 1 1 ) n n 1 n e. i=1

24 Counting Regular Hypergraphs by Girth r-regular hypergraph: Every vertex belongs to exactly r edges girth g: The hypergraph contains no k-cycle for k < g

25 Counting Regular Hypergraphs by Girth A typical k-cycle in a hypergraph is one in which consecutive edges intersect in exactly one vertex. 3-cycle in a 3-uniform hypergraph

26 Counting Regular Hypergraphs by Girth The configuration model [Bollobás 1980] connects matchings with multihypergraphs. Matching Graph 2-regular, 2-uniform hypergraph (i.e. graph) on three vertices

27 Counting Regular Hypergraphs by Girth The configuration model [Bollobás 1980] connects matchings with multihypergraphs. Matching Multigraph 3-regular, 2-uniform multihypergraph (i.e. multigraph) on four vertices

28 Counting Regular Hypergraphs by Girth Let M contain all matchings whose projection is a k-cycle with k < g. For such a collection, the set M M A M contains precisely the matchings that represent hypergraphs of girth at least g in the configuration model.

29 Counting Regular Hypergraphs by Girth Theorem (M 2013) In the configuration model, assume g 3, r 3 and (2r 2) 2g 3 g 3 = o(n). The number of simple r-regular, 3-uniform hypergraphs on N vertices with girth at least g is ( ) g 1 (2r 2) i (1 + o(1)) exp 2i i=1 6 rn/3 ( rn 3 (rn)! ).!(r!) N

30 Thanks Contact Me Download This Talk AustinMohr.com Further Details Lu and Székely, 2009 A New Asymptotic Enumeration Technique: The Lovász Local Lemma Lu, M, and Székely, 2012 Quest for Negative Dependency Graphs