Applications of the Lopsided Lovász Local Lemma Regarding Hypergraphs

Transcription

1 Regarding Hypergraphs Ph.D. Dissertation Defense April 15, 2013

2 Overview The Local Lemmata 2-Coloring Hypergraphs with the Original Local Lemma Counting Derangements with the Lopsided Local Lemma Lopsided Spaces Uniform Hypergraph Matchings Set Partitions Spanning Trees Applications Enumeration of Regular, Uniform Hypergraphs Characterization of Perfect Matching Hosts

3 Note on Probability Spaces For this talk, every a probability space Ω is assumed to be uniform and equipped with the counting measure, so that for any subset A of Ω. Pr (A) = A Ω

4 The Local Lemmata The Local Lemmata

5 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.

6 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 }}

7 2-Coloring Hypergraphs A 2-coloring is an assignment of two colors to the vertices and is proper if no edge is monochromatic.

8 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?

9 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

10 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.

11 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}

12 2-Coloring Hypergraphs Graph to be 2-colored Dependency Graph

13 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

14 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.

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 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.

18 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.

19 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.

20 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.

21 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].

22 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.

23 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.

24 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]}.

25 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

26 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

27 Lopsided Spaces Lopsided Spaces

28 Negative Dependency Graph for Matchings s-uniform hypergraph: Every edge contains exactly s vertices (partial) matching: Collection of vertex-disjoint edges maximal matching: Not contained by a strictly larger matching

29 Negative Dependency Graph for Matchings The canonical event for a matching is the collection of all maximal matchings extending it. partial matching B canonical event A B

30 Negative Dependency Graph for Matchings Conflict Graph Each vertex corresponds to a matching. Two matchings are adjacent if their union is not again a partial matching.

31 Negative Dependency Graph for Matchings Conflict Graph Each vertex corresponds to a matching. Two matchings are adjacent if their union is not again a partial matching. Theorem (Lu, M, Székely 2013) Let M be any collection of matchings in a complete uniform hypergraph. The conflict graph is a negative dependency graph for the canonical events {A M M M}.

32 Negative Dependency Graph for Partitions The canonical event for a partial partition is the collection of all partitions extending it. partial partition P of [6] canonical event A P

33 Negative Dependency Graph for Partitions Conflict Graph Each vertex of corresponds to a partial partition. Two partial partitions are adjacent if their union is not again a partial partition.

34 Negative Dependency Graph for Partitions The conflict graph for the partial partitions 2 3, 1 3, 1 2, and 4 is not a negative dependency graph.

35 Negative Dependency Graph for Partitions The conflict graph for sufficiently coarse partial partitions is a negative dependency graph, however. For a given collection P of partial partitions, define P P a (P) = P, P the average number of blocks in a partial partition belonging to P.

36 Negative Dependency Graph for Partitions One can show where a(ω N ) = B N+1 B N 1, Ω N is the collection of all partitions of [N] and the Nth Bell number B N counts the number of partitions of [N] (i.e. B N = Ω N ).

37 Negative Dependency Graph for Partitions Theorem (M 2013) Let P be a collection of partial partitions of a large set [N]. If ( ) a A P B N+1 1, B N P P then the conflict graph is a negative dependency graph for the canonical events {A P P P}. Corollary (M 2013) Let P be a collection of partial partitions of a large set [N]. If every block appearing in P has size at least log N, then the conflict graph is a negative dependency graph for the canonical events {A P P P}.

38 Negative Dependency Graph for Partitions The proof of the corollary makes use of a powerful asymptotic expression for the Bell numbers. Lemma (Canfield 1995) Let r be the unique real solution of the equation re r = N. The identity (N + h)! B N+h = r N+h exp ( (er 1) (2πB) 1/2 1 + P e r + Q ) e 2r + O(e 3r ) holds uniformly for h = O(log N) as N tends to infinity, where B is exponential in r and P and Q are rational functions in r and h.

39 Negative Dependency Graph for Spanning Trees forest: A cycle-free graph tree: A connected forest spanning tree of K N : A tree T with V (T ) = V (K N )

40 Negative Dependency Graph for Spanning Trees The canonical event for a forest in K N is the collection of all spanning trees of K N containing it. forest F of K 4

41 Negative Dependency Graph for Spanning Trees The canonical event for a forest in K N is the collection of all spanning trees of K N containing it. canonical event A F

42 Negative Dependency Graph for Spanning Trees Conflict Graph Each vertex corresponds to a forest. Two forests are adjacent if they possess components that are neither identical nor disjoint.

43 Negative Dependency Graph for Spanning Trees Conflict Graph Each vertex corresponds to a forest. Two forests are adjacent if they possess components that are neither identical nor disjoint. Theorem (Lu, M, Székely 2013) Let F be any collection of forests in a complete graph. The conflict graph is a negative dependency graph for the canonical events {A F F F}.

44 Negative Dependency Graph for Spanning Trees The theorem relies on an extension of Cayley s theorem for spanning trees. Theorem (Cayley 1889) There are N N 2 spanning trees of K N. Theorem (Lu, M, Székely 2013) Let F be a forest in K N with connected components T 1,..., T k on t 1,..., t k vertices, respectively. There are N N 2 k i=1 spanning trees of K N containing F. t i N t i 1 If F contains no edges, then the extension gives back Cayley s result.

45 Applications Applications

46 Asymptotics from the Lopsided Local Lemma ɛ-near Positive Dependency Graph G [Lu, Székely 2011] Each vertex of G corresponds to an event. Pr (A i A j ) = 0 whenever ij E(G). For each event A i and any subset S of its non-neighbors in G, Pr A i (1 ɛ) Pr (A i ) j S A j Theorem (M 2013) Let M be a collection of matchings in a complete s-uniform hypergraph. If M is sufficiently sparse, then the conflict graph for the canonical events {A M M M} is also an ɛ-near positive dependency graph.

47 Asymptotics from the Lopsided Local Lemma Let A 1,..., A n be events in a probability space Ω N that grows with N and set µ = n i=1 Pr (A i). If the probabilities are appropriately controlled, then a negative dependency graph gives the lower bound ( n ) Pr A i (1 o(1))e µ [Lu, Székely 2011] i=1 and a positive dependency graph gives the upper bound ( n ) Pr A i (1 + o(1))e µ [M 2013] i=1 as N tends to infinity.

48 Asymptotics from the Lopsided Local Lemma Corollary (M 2013) Let A 1,..., A n be events in a probability space Ω N. If the conditions of the previous two theorems are satisfied, then n A i = (1 + o(1))e µ Ω N i=1 as N tends to infinity.

49 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

50 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

51 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

52 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

53 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.

54 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

55 Counting Regular Hypergraphs by Girth Conjecture (M 2013) In the configuration model, assume g 3, r 3, s 2, and (s 1) 2g 3 (r 1) 2g 3 g 3 = o(n). The number of simple r-regular, s-uniform hypergraphs on N vertices with girth at least g is ( ) g 1 (s 1) i (r 1) i (1 + o(1)) exp 2i i=1 (s!) rn/s ( rn s (rn)! ).!(r!) N The conjecture has been proven for s {2, 3}.

56 Counting Partitions by Block Size Conjecture (M 2013) The number of partitions of [N] whose smallest block is of size m is (1 + o(1)) exp ( m 1 i=1 ( N i ) BN i B N ) B N (assuming restricted growth of m as a function of N). The conjecture has been proven for m = 2, the number of singleton-free partitions.

57 Perfect Matching Hosts Redefine the canonical event for a matching in a hypergraph to be the collection of all perfect matchings containing it. A perfect matching host is a hypergraph H for which the following statement is true: Let M be any collection of matchings in H. The conflict graph is a negative dependency graph for the canonical events {A M M M}.

58 Perfect Matching Hosts A hypergraph is k-randomly matchable provided every partial matching containing at most k edges can be extended to a perfect matching. A randomly matchable hypergraph is one that is k-randomly matchable for all k. Lemma (M 2013) If a perfect matching host is 1-randomly matchable, then it is randomly matchable.

59 Perfect Matching Hosts A perfect matching host is therefore the union of a randomly matchable perfect matching host together with edges that belong to no perfect matching. Theorem (M 2013) A hypergraph H is a perfect matching host if and only if H[A] is a (possibly disconnected) randomly matchable perfect matching host, where A contains all edges of H that belong to at least one perfect matching.

60 Perfect Matching Hosts The randomly matchable 2-uniform hypergraphs (i.e. graphs) have been characterized completely. Theorem (Sumner 1979) The only randomly matchable graphs are even cliques and balanced bicliques.

61 Perfect Matching Hosts Theorem (M 2013) A graph G is a perfect matching host if and only if there is a partition of the edges into sets A and B such that H[A] is a disjoint union of even cliques and balanced bicliques and there is no subset F of the edges of B such that F has an even number of vertices in common with each even clique of H[A], and for any balanced biclique of G[A], F has an equal number of vertices in common with both of its partite sets. Perfect matching host

62 Perfect Matching Hosts Theorem (M 2013) A graph G is a perfect matching host if and only if there is a partition of the edges into sets A and B such that H[A] is a disjoint union of even cliques and balanced bicliques and there is no subset F of the edges of B such that F has an even number of vertices in common with each even clique of H[A], and for any balanced biclique of G[A], F has an equal number of vertices in common with both of its partite sets. Not a perfect matching host (evidenced by red edges)

63 Perfect Matching Hosts Matching hosts for 3-uniform hypergraphs appear to be much more exotic.

64 Thanks Further Information L. Lu, A. Mohr, and L. A. Székely, Quest for Negative Dependency Graphs, Recent Advances in Harmonic Analysis and Applications (in honor of Konstantin Oskolkov), Springer Proceedings in Mathematics and Statistics 25 (2013). L. Lu, A. Mohr, L. A. Székely, Connected Balanced Subgraphs in Random Regular Multigraphs Under the Configuration Model, Journal of Combinatorial Mathematics and Combinatorial Computing (2013+). At least two more in preparation.