Lovász Local Lemma a new tool to asymptotic enumeration?

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1 Lovász Local Lemma a ew tool to asymptotic eumeratio? Liyua Licol Lu László Székely Uiversity of South Carolia Supported by NSF DMS

2 Overview LLL ad its geeralizatios LLL a istace of the Poisso paradigm New egative depedecy graphs Applicatios: Permutatio eumeratio Lati rectagle eumeratio Regular graph eumeratio A joke

3 Whe oe of the evets happe Assume that A 1,A 2,,A are evets i a probability space Ω. How ca we ifer A i? i=1 If A i s are mutually idepedet, P(A i )<1, the P A i i=1 = P ( A ) i = ( 1 P( A i )) > 0 i=1 i=1 If ( ), the P A i < 1 i=1 P i=1 A i = P i=1 A i = 1 P i=1 A i 1 ( ) P A i > 0 i=1

4 A way to combie argumets: Assume that A 1,A 2,,A are evets i a probability space Ω. Graph G is a depedecy graph of the evets A 1,A 2,,A, if V(G)={1,2,,} ad each A i is idepedet of the elemets of the evet algebra geerated by { A j :ij E( G) }

5 Lovász Local Lemma (Erdős-Lovász 1975) Assume G is a depedecy graph for A 1,A 2,,A, ad d=max degree i G If for i=1,2,,, P(A i )<p, ad e(d+1)p<1, the P i=1 A i > 0

6 Lovász Local Lemma (Specer) Assume G is a depedecy graph for A 1,A 2,,A If there exist x 1,x 2,,x i [0,1) such that the P ( ) x i 1 x j P A i i=1 A i ( ) ij E G ( ) ( ) 1 x i > 0 i=1

7 Negative depedecy graphs Assume that A 1,A 2,,A are evets i a probability space Ω. Graph G with V(G)={1,2,,} is a egative depedecy graph for evets A 1,A 2,,A, if i S { j :ij E( G) } P j S A j implies > 0 P A i A j j S P A i ( )

8 LLL: Erdős-Specer 1991, Albert-Freeze-Reed 1995, Ku Assume G is a egative depedecy graph for A 1,A 2,,A, exist x 1,x 2,,x i [0,1) such that, P( A i ) x ( i 1 x ) j, the P i=1 A i ( ) ( ) ij E G 1 x i > 0 i=1 Settig x i =1/(d+1) implies the uiform versio both for depedecy ad egative depedecy

9 Needle i the haystack LLL has bee i use for existece proofs to exhibit the existece of evets of tiy probability. Is it good for other purposes? Where to fid egative depedecy graphs that are ot depedecy graphs?

10 Poisso paradigm Assume that A 1,A 2,,A are evets i a probability space Ω, p(a i )=p i. Let X deote the sum of idicator variables of the evets. If depedecies are rare, X ca be approximated with Poisso distributio of mea Σ p i. X~Poisso meas usig k=0, P i=1 A i ( ) = e µ µ k / k! P X = k e µ = e i=1 p i

11 Models for the Poisso paradigm Che-Stei method Jaso iequality 1990 Bru s sieve Now: LLL. Assume G is egative depedecy graph, 0<ε<0.14. i : P A i ( ) < ε; P A j 1+ 3ε ( ) < ε imply P A j e j=1 ( ) ij E G j =1 ( ) ( ) P A j

12 Two egative depedecy graphs H is a complete graph K N or a complete bipartite graph K N,L ; Ω is the uiform probability space of maximal matchigs i H. For a partial matchig M, the caoical evet A M = F Ω M F { } Caoical evets A M ad A M* are i coflict: M ad M* have o commo extesio ito maximal matchig, i.e. A M A M * =

13 Mai theorem For a graph H=K N or K N,L, ad a family of caoical evets, if the edges of the graph G are defied by coflicts, the G is a egative depedecy graph. This theorem fails to exted for the hexago H=C 6

14 Hexago example Two perfect matchigs e p( A ) e = p( A ) f = 1 2 f ( ) = p A e A f p A e A f ( ) p( A ) f = 1 2 = 1 p( A ) 1 e 2

15 Relevace for permutatio eumeratio problems Deragemets avoid: 2-cycle free avoids: 3-cycle free avoids: i i i j i j i j k i j k

16 ε-ear-positive depedecy graphs Assume that A 1,A 2,,A are evets i a probability space Ω. Graph G with V(G)={1,2,,} is a ε ear-positive depedecy graph of the evets A 1,A 2,,A, ( ) = 0 ij E(G) implies P A i A j i S { j :ij E( G) } P A j j S > 0 implies P A i j S A j ( 1 ε)p A i ( )

17 Quotiet graphs Assume G is a egative depedecy graph for A 1,A 2,,A. Assume further that V(G) is partitioed ito classes such that evets i the same class are disjoit. For every partitio class J, let B J = A j. The quotiet graph of G is a j J egative depedecy graph for the evets B J

18 Quotiet graphs of ε-ear-positive depedecy graphs If the oly edges of the quotiet graph of a ε ear-positive depedecy graph are loops, the the quotiet graph is also a ε-ear-positive depedecy graph.

19 Asymptotic results A collectio of matchigs M is regular, if for every i, every vertex is covered d i times by i-elemet matchigs from M A collectio of matchigs M is δ-sparse (details avoided!) Negative depedecy graphs of δ- sparse collectios of matchigs are also ε-ear-positive depedecy graphs

20 Asymptotic results a theorem A collectio of matchigs M i K N or K N,N is regular, r is the largest matchig size, M is δ-sparse. Set µ = P( A ) M over M. Suppose δ=o(µ -1 ), µ is separated from 0, ad r = o( N ) The P M A M ( ) µ = o Nr 3/2 = ( 1 + o(1) )e µ

21 Cosequeces for permutatio eumeratio For k fixed, the proportio of k-cycle free permutatios is ( 1 o( 1) )e 1 k (Beder 70 s) If max K grows slowly with, the proportio of permutatios free of cycles of legth from set K is ( 1 o( 1) )e k K 1 k

22 Lati rectagles Lati rectagle: k times array filled with etries 1,2,,; puttig a permutatio ito every row ad ot repeatig a etry i ay colum (k ) L(k,)= umber of k times Lati rectagles

23 Eumeratio of Lati rectagles L(2,)=! (# of deragemets) (!) 2 e -1 Riorda 1944 L(3,) (!) 3 e -3 Erdős-Kaplasky 1946 L( k,) (! ) k e k 2 for k = o (( log) 3 ) 2 Yamamoto 1951 exteded to ( ) k = o 1 3 ε

24 Eumeratio of Lati rectagles Stei 1978 (usig Che-Stei method) L( k,) (! ) k e k 2 k3 6 ( ) for k = o 1 2 Godsil ad McKay 1990 refied the asymptotics to make it work for ( ) k = o 6 7

25 Eumeratio of Lati rectagles Skau 1990 (usig va der Waerde s iequality for the permaet) (! ) k k 1 1 r r =1 ad with this matched Stei s lower boud o a slightly smaller rage: L( k,) ( 1 o(1) )(! ) k e L( k,) k 2 k3 6 ( for k = o 1 2 ) log

26 Eumeratio of Lati rectagles Quotiet graph versio of the egative depedecy graph LLL yields Skau s lower boud: matches the rage of Stei s lower boud: L k, (! ) k k 1 1 r r =1 ( ) ( 1 o(1) )(! ) k e L( k,) k 2 k3 6 ( for k = o 1 2 ) log

27 Eumeratio of Lati rectagles Quotiet graph versio of the ear ε- positive depedecy graph argumet yields tight asymptotic upper boud i Yamamoto s rage: L( k,) ( 1 o(1) )(! ) k e k 2 for k = o( 1 3 ε )

28 Relevace for Lati rectagle eumeratio Try to fill i the fourth row with a permutatio of [5]. Complete bipartite graph: 1st class colums, 2d class etries Caoical evets defied by the edges 11, 13, 14; 23, 22, 25; 34, 35, 31; 42, 44, 43; 55, 51, 52

29 Eumeratio of labeled regular graphs Beder-Cafield, idepedetly Wormald 1978: d fix, d eve ( 2e 1 d2 )/4 d d d e d ( d! ) 2 2

30 Cofiguratio model (Bollobás 1980) Put d (d eve) vertices ito equal clusters Pick a radom matchig of K d Cotract every cluster ito a sigle vertex gettig a multigraph or a simple graph Observe that all simple graphs are equiprobable

31 Eumeratio of labeled regular graphs Bollobás 1980: d eve, d < 2log ( 1+ o( 1) )e 1 d2 ( ( )/4 d 1)!! ( d! ) McKay 1985: for ( d = o 1 ) 3

32 Eumeratio of labeled regular graphs McKay ad Wormald: d eve, ( ( 1 + o( 1) )e 1 d2 )/4 d 3 /( 12)+O d 2 / ( ) d 1 Wormald 1981: fix d 3, g 3 girth ( 1+ o( 1) )e g 1 ( d 1) i ( 2i d 1)!! i=1 ( d! ) d = o 1 2 ( )!! ( d! ) ( )

33 Theorem I the cofiguratio model, if d 3 ad g 3 d 2g-3 =o(), the the probability that the resultig radom d-regular multigraph after the cotractio has girth at least g, is ( 1 + o( 1) )e hece the umber of d-regular graphs with girth at least g is ( 1+ o( 1) )e g 1 i=1 g 1 i=1 ( d 1) i ( d 1) i 2i ( 2i d 1)!! ( d! )

34 Specer s joke Specer s joke i Te Lectures o the Probabilistic Method : uiformly selected radom fuctio from [a] to [b] is ijectio whp, if b>>a 2, but: LLL ca show the existece of a ijectio whe 2ea < b.

35 Our joke Our joke: usig egative depedecy graph LLL, a uiformly selected radom fuctio [a] to [a] is ijectio with probability at least a! a a > 1 e o(1) Combiatorial proof to the slightly weakeed Stirlig formula a

36 Symmetric evets Evets A 1,A 2,,A are symmetric, if the probability of their Boolea expressios do ot chage, whe A π(i) is substituted for A i simultaeously for ay permutatio π.

37 A Lemma for symmetric evets If evets A 1,A 2,,A are symmetric p i = P i j =1 A j ad p = 1, 0 i = 1,2,..., 1 p i 2 p i 1 p i+1 the LLL applies with empty egative depedecy graph, x i =p 1

38 Proof to our joke Cosider a uiform radom fuctio f from [a] to [b] with a b. Let A u deote the evet that f(u) occurs with multiplicity 2 or higher. P( A ) u = 1 b ( ) 1 P A 1 P ij (b 1)a 1 ad p b a i = i! ( ( ) ) a = 1 1 a a(a 1) b i (b i) a i b a = 1 e o(1) a

Connected Balanced Subgraphs in Random Regular Multigraphs Under the Configuration Model

Connected Balanced Subgraphs in Random Regular Multigraphs Under the Configuration Model Coected Balaced Subgraphs i Radom Regular Multigraphs Uder the Cofiguratio Model Liyua Lu Austi Mohr László Székely Departmet of Mathematics Uiversity of South Carolia 1523 Greee Street, Columbia, SC 29208