The Lopsided Lovasz Local Lemma
Austin Mohr
Department of MathematicsNebraska Wesleyan University
With Linyuan Lu and Laszlo Szekely, University of South Carolina
Austin Mohr The Lopsided Lovasz Local Lemma
Note on Probability Spaces
For this talk, every a probability space Ω is assumed to be uniformand equipped with the counting measure, so that
Pr (A) =|A||Ω|
for any subset A of Ω.
Austin Mohr The Lopsided Lovasz Local Lemma
Lovasz Local Lemma
Given a collection of mostly independent bad events, there is a wayto avoid them all.
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
Hypergraph
Ground set V
Collection E of nonempty subsets of V
V = vi | i ∈ [5]E = v1, v2, v3, v2, v3, v3, v4, v5
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
A 2-coloring is an assignment of two colors to the vertices and isproper if no edge is monochromatic.
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
If there are many intersections among small edges, a hypergraphmay be impossible to 2-color properly.
How many intersections can we allow and still guarantee a proper2-coloring exists?
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
Color the vertices of a hypergraph H with two colors independentlyat random.
For each edge e, define the “bad” event
Ae = 2-colorings of H | e is monochromatic.
The proper 2-colorability of H is equivalent to
Pr
∧e∈E(H)
Ae
> 0.
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
Let e be an edge of the hypergraph H and F be a collection ofedges that are disjoint from e.
The event Ae is independent of the event algebra generated byAf | f ∈ F.
We capture this information in a dependency graph.
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
Dependency Graph G [Erdos, Lovasz 1975]
Each vertex corresponds to an event.
Each event is independent of the event algebra generated byits non-neighbors in G .
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
For hypergraph 2-coloring, the graph G with
V (G ) = E (H) and
E (G ) = ef | e and f share a vertex in H
is a dependency graph.
Hypergraph to be 2-colored Dependency Graph
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
Lemma (Symmetric Local Lemma - Erdos, Lovasz 1975)
Let Ai | i ∈ [n] be a collection of events having a dependencygraph G such that
G has maximum degree d and
Pr (Ai ) ≤ p for all i .
If ep(d + 1) ≤ 1, then
Pr
(n∧
i=1
Ai
)> 0.
Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
For the hypergraph 2-coloring problem,
Pr (Ai ) ≤ 22k
= p (k is the size of the smallest edge) and
d is the greatest number of intersections witnessed by anyedge.
The local lemma requires
ep(d + 1) ≤ 1
and so
d ≤ 2k−1
e− 1.
With d bounded, the local lemma concludes
Pr
(n∧
i=1
Ai
)> 0,
which means it is possible to properly 2-color the hypergraph.Austin Mohr The Lopsided Lovasz Local Lemma
2-Coloring Hypergraphs
Theorem (Erdos, Lovasz 1975)
Let H be a hypergraph in which every edge contains at least kvertices. If each edge intersects at most 2k−1
e − 1 other edges, thenH is properly 2-colorable.
Austin Mohr The Lopsided Lovasz Local Lemma
Lovasz Local Lemma
Lemma (Asymmetric Local Lemma - Spencer 1977)
Let Ai | i ∈ [n] be a collection of events having a dependencygraph G .If there are real numbers xi ∈ [0, 1) such that
Pr (Ai ) ≤ xi∏
ij∈E(G)
(1− xj)
for all i , then
Pr
(n∧
i=1
Ai
)≥
n∏i=1
(1− xi ) > 0.
The symmetric version follows from the asymmetric version bysetting each xi = 1
d+1 .
Austin Mohr The Lopsided Lovasz Local Lemma
Derangements
A derangement is a permutation of [n] having no fixed point.
Define the canonical event
Ai = permutations π of [n] | π(i) = i .
The set⋂n
i=1 Ai contains precisely the derangements of [n].
Austin Mohr The Lopsided Lovasz Local Lemma
Derangements
The local lemma does not apply, since no pair of the events areindependent:
Pr (A1 ∧ A2) =(n − 2)!
n!=
1
n2 − n,
while
Pr (A1) Pr (A2) =(n − 1)!
n!· (n − 1)!
n!=
1
n2.
Austin Mohr The Lopsided Lovasz Local Lemma
Derangements
Fortunately, derangements possess a different useful property:
Pr (A1) =(n − 1)!
n!=
1
n
and
Pr(A1
∣∣ A2
)=|A1 ∩ A2||A2|
=(n − 1)!− (n − 2)!
n!− (n − 1)!=
1
n− 1
n(n − 1)2
soPr(A1
∣∣ A2
)≤ Pr (A1)
Austin Mohr The Lopsided Lovasz Local Lemma
Derangements
In fact,
Pr
(A1
∣∣∣∣∣k∧
i=2
Ai
)≤ Pr (A1)
for any k .
We capture this information in a negative dependency graph.
Austin Mohr The Lopsided Lovasz Local Lemma
Derangements
Negative Dependency Graph G [Erdos, Spencer 1991]
Each vertex corresponds to an event.
The inequality
Pr
Ai
∣∣∣∣∣∣∧j∈S
Aj
≤ Pr (Ai )
holds for each event Ai and any subset S of its non-neighborsin G .
Theorem (Lu, Szekely 2006)
The graph with vertex set [n] and no edges is a negativedependency graph for the canonical events Ai | i ∈ [n].
Austin Mohr The Lopsided Lovasz Local Lemma
Lopsided Local Lemma
Given a collection of mostly negative dependent bad events, thereis a way to avoid them all.
Lemma (Lopsided Local Lemma - Erdos, Spencer 1991)
Let Ai | i ∈ [n] be a collection of events having negativedependency graph G .If there are real numbers xi ∈ [0, 1) such that
Pr (Ai ) ≤ xi∏
ij∈E(G)
(1− xj)
for all i , then
Pr
(n∧
i=1
Ai
)≥
n∏i=1
(1− xi ) > 0.
Austin Mohr The Lopsided Lovasz Local Lemma
Derangements
Setting each xi = 1n , we verify
Pr (Ai ) ≤1
n
∏ij∈∅
(1− 1
n
)
and conclude
Pr
(n∧
i=1
Ai
)≥(
1− 1
n
)nn−→ 1
e.
Austin Mohr The Lopsided Lovasz Local Lemma
Hypergraph Matchings
The canonical event for a partial matching is the collection of allperfect matchings extending it.
partial matching B
canonical event AB
Austin Mohr The Lopsided Lovasz Local Lemma
Hypergraph Matchings
Conflict Graph
Each vertex corresponds to a partial matching.
Two matchings are adjacent if their union is not again apartial matching.
Austin Mohr The Lopsided Lovasz Local Lemma
Hypergraph Matchings
Theorem (Lu, M, Szekely 2013)
Let M be any collection of matchings in a complete uniformhypergraph. The conflict graph is a negative dependency graph forthe canonical events AM | M ∈M.
The set ⋂M∈M
AM
contains all perfect matchings of the complete uniform hypergraphthat extend no matching from M.
Austin Mohr The Lopsided Lovasz Local Lemma
Asymptotics from the Lopsided Local Lemma
ε-Near Positive Dependency Graph G [Lu, Szekely 2011]
Each vertex of G corresponds to an event.
Pr (Ai ∧ Aj) = 0 whenever ij ∈ E (G ).
The inequality
Pr
Ai
∣∣∣∣∣∣∧j∈S
Aj
≥ (1− ε) Pr (Ai )
holds for each event Ai and any subset S of its non-neighborsin G .
Theorem (M 2013)
Let M be a collection of matchings in a complete uniformhypergraph. If M is sufficiently “sparse”, then the conflict graphfor the canonical events AM | M ∈M is an ε-near positivedependency graph.
Austin Mohr The Lopsided Lovasz Local Lemma
Asymptotics from the Lopsided Local Lemma
Let A1, . . . , An be events in a probability space ΩN that growswith N and set µ =
∑ni=1 Pr (Ai ).
If the probabilities are appropriately controlled, then a negativedependency graph gives the lower bound
Pr
(n∧
i=1
Ai
)≥ (1− o(1))e−µ [Lu, Szekely 2011]
and a positive dependency graph gives the upper bound
Pr
(n∧
i=1
Ai
)≤ (1 + o(1))e−µ [M 2013]
as N tends to infinity.
Austin Mohr The Lopsided Lovasz Local Lemma
Asymptotics from the Lopsided Local Lemma
Corollary (M 2013)
Let A1, . . . , An be events in a probability space ΩN . If theconditions of the previous two theorems are satisfied, then∣∣∣∣∣
n⋂i=1
Ai
∣∣∣∣∣ = (1 + o(1))e−µ|ΩN |
as N tends to infinity.
Austin Mohr The Lopsided Lovasz Local Lemma
Counting Hypergraphs by Girth
A typical k-cycle in a hypergraph is one in which consecutiveedges intersect in exactly one vertex.
Figure : 3-cycle in 4-uniform hypergraph.
Austin Mohr The Lopsided Lovasz Local Lemma
Counting Hypergraphs by Girth
The configuration model [Bollobas 1980] connects matchingswith multihypergraphs.
2-Matching Multihypergraph
Figure : Configuration projecting to 3-regular, 2-uniform multihypergraphon four vertices.
Austin Mohr The Lopsided Lovasz Local Lemma
Counting Hypergraphs by Girth
Let M contain all matchings whose projection is a k-cycle withk < g .
For such a collection, the set ⋂M∈M
AM
contains precisely the matchings that represent hypergraphs ofgirth at least g in the configuration model.
Austin Mohr The Lopsided Lovasz Local Lemma
Counting Hypergraphs by Girth
Theorem (M, Szekely 2013+)
The number of r -regular, s-uniform hypergraphs having girth atleast g is
(1 + o(1))(rN)!
s!rN/s(rNs
)!(r !)N
exp
(−
g−1∑i=1
(r − 1)i (s − 1)i
2i
)
(assuming g, r , and s grow slowly with N).
Austin Mohr The Lopsided Lovasz Local Lemma
Homework
1 Think of your favorite combinatorial object.
Partial matchings
2 Define the canonical event for a particlar instance of thattype.
All perfect matchings extending it
3 Define conflict for two objects of that type.
Union is not a matching
4 Determine whether your conflict graph is a negativedependency graph.
Ask Laszlo
Austin Mohr The Lopsided Lovasz Local Lemma
Homework
1 Think of your favorite combinatorial object.
Partial matchings
2 Define the canonical event for a particlar instance of thattype.
All perfect matchings extending it
3 Define conflict for two objects of that type.
Union is not a matching
4 Determine whether your conflict graph is a negativedependency graph.
Ask Laszlo
Austin Mohr The Lopsided Lovasz Local Lemma
Homework
1 Think of your favorite combinatorial object.
Partial matchings
2 Define the canonical event for a particlar instance of thattype.
All perfect matchings extending it
3 Define conflict for two objects of that type.
Union is not a matching
4 Determine whether your conflict graph is a negativedependency graph.
Ask Laszlo
Austin Mohr The Lopsided Lovasz Local Lemma
Homework
1 Think of your favorite combinatorial object.
Partial matchings
2 Define the canonical event for a particlar instance of thattype.
All perfect matchings extending it
3 Define conflict for two objects of that type.
Union is not a matching
4 Determine whether your conflict graph is a negativedependency graph.
Ask Laszlo
Austin Mohr The Lopsided Lovasz Local Lemma
Thanks
Download This Talkwww.AustinMohr.com
Further Information
A. Mohr, Applications of the Lopsided Lovasz Local LemmaRegarding Hypergraphs, Ph.D. Dissertation (2013).
L. Lu, A. Mohr, and L. A. Szekely, Quest for NegativeDependency Graphs, Recent Advances in Harmonic Analysisand Applications (in honor of Konstantin Oskolkov), SpringerProceedings in Mathematics and Statistics 25 (2013).
L. Lu, A. Mohr, and L. A. Szekely, Connected BalancedSubgraphs in Random Regular Multigraphs Under theConfiguration Model , Journal of Combinatorial Mathematicsand Combinatorial Computing (2013+).
Austin Mohr The Lopsided Lovasz Local Lemma