Chapter 4: The Lovsz Local Lemma The Probabilistic Method Summer - PowerPoint PPT Presentation

Chapter 4: The Lovász Local Lemma The Probabilistic Method Summer 2020 Freie Universität Berlin

Chapter Overview • Introduce the Lovász Local Lemma and some variants • Survey some applications, including to 𝑆 3, 𝑙

§1 Introducing the Lemma Chapter 4: The Lovász Local Lemma The Probabilistic Method

Avoiding Bad Events Second moment set-up • Have a collection of good events 𝐹 1 , 𝐹 2 , … , 𝐹 𝑛 • e.g.: 𝐹 𝑗 = 𝑗th copy of 𝐼 appears in 𝐻 𝑜, 𝑞 • Goal: show that with positive probability, at least one event occurs • Usually show this happens with probability 1 − 𝑝 1 Opposite situation • Have a collection of bad events 𝐹 1 , 𝐹 2 , … , 𝐹 𝑛 • e.g.: 𝐹 𝑗 = 𝑗th clause in 𝑙−SAT formula not satisfied • Goal: show that with positive probability, none of these events occur 𝑑 > 0 • i.e.: ℙ ∩ 𝑗 𝐹 𝑗 𝑑 = 1 − ℙ ∪ 𝑗 𝐹 𝑗 ≥ 1 − σ 𝑗 ℙ 𝐹 𝑗 • Union bound: ℙ ∩ 𝑗 𝐹 𝑗 • Tight when events 𝐹 𝑗 are disjoint • In general, need either 𝑛 or ℙ 𝐹 𝑗 to be small enough for effective bounds

Independence to the Rescue Independent events • If the events 𝐹 1 , 𝐹 2 , … , 𝐹 𝑛 are mutually independent, we are in business 𝑑 = ς 𝑗 ℙ 𝐹 𝑗 𝑑 = ς 𝑗 1 − ℙ 𝐹 𝑗 • ℙ ∩ 𝑗 𝐹 𝑗 • Might tend to zero, but is still positive (provided ℙ 𝐹 𝑗 < 1 for all 𝑗 ) • Doesn’t matter how many bad events there are, or how likely they are A real-world example • Work for the Bundesdruckerei • Job: printing 𝑛 passports • Bad event: 𝐹 𝑗 = misprint in the 𝑗th passport 1 • Say ℙ 𝐹 𝑗 = 2 for each 𝑗 𝑛 𝑑 = 1 • ℙ ∩ 𝑗 𝐹 𝑗 > 0 2 • ⇒ it is possible to have a successful day

The Struggle for Independence Do we need independence? • In practice, true independence of events is rare • Could hope to replace it with something weaker • Most events being independent? Pairwise independence? We might • Bundesdruckerei example: suppose our passport printer is odd • Never makes an even number of misprints • Same marginal distributions 1 • ℙ 𝐹 𝑗 = 2 for all 𝑗 • Almost complete independence • Any 𝑛 − 1 of the 𝑛 events are mutually independent 𝑑 ≤ ℙ # misprints even = 0 • However, ℙ ∩ 𝑗 𝐹 𝑗

Lovász to the Rescue The Bundesdruckerei problem 1 • ℙ 𝐹 𝑗 = 2 is a large probability for the bad event 1 • If ℙ 𝐹 𝑗 < 2 , then we lose even pairwise independence • ℙ 𝐹 𝑗 𝐹 𝑘 < ℙ 𝐹 𝑗 The good news • Suppose the bad events • are independent of most other events • occur with reasonably small probability • Lovász Local Lemma ⇒ events behave as if independent • Can show that with positive probability none occur

The Local Lemma – Symmetric Setting Theorem 4.1.1 (Symmetric Lovász Local Lemma; Erd ő s-Lovász, 1975) Let 𝐹 1 , 𝐹 2 , … , 𝐹 𝑛 be events such that each event 𝐹 𝑗 is mutually independent of all but at most 𝑒 of the other events, and ℙ 𝐹 𝑗 ≤ 𝑞 𝑑 > 0 . for all 𝑗 . If 𝑓𝑞 𝑒 + 1 ≤ 1 , then ℙ ∩ 𝑗 𝐹 𝑗 “Local” Lemma • Bound on 𝑞 independent of number of events (global property) • Only depends on number of dependencies (local property) Conclusion • Only assert that with positive probability, none of the events occur • This probability can depend on the number of events

Re-restricted 𝑙 -SAT Recall • Any 𝑙 -SAT formula with fewer than 2 𝑙 clauses is satisfiable • Bound is best possible: take formula with all clauses on 𝑙 variables Restricted 𝑙 -SAT • Previously: each 𝑙 -set of variables appears in at most one clause • What if we bound individual variable appearances instead? Theorem 4.1.2 2 𝑙 Any 𝑙 -SAT formula in which each variable appears at most 𝑓𝑙 times is satisfiable. • Applies to 𝑙 -SAT formulae with any number of clauses!

Proof by Local Lemma Theorem 4.1.2 2 𝑙 Any 𝑙 -SAT formula in which each variable appears at most 𝑓𝑙 times is satisfiable. Proof 1 • Set each variable to true/false independently with probability 2 • Events • 𝐹 𝑗 = 𝑗th clause not satisfied • ℙ 𝐹 𝑗 = 𝑞 ≔ 2 −𝑙 for all 𝑗 • Dependencies • A clause is independent of any clauses with disjoint sets of variables 2 𝑙 • Clause has 𝑙 variables, each in ≤ 𝑓𝑙 − 1 other clauses 2 𝑙 • ⇒ each event independent of all but 𝑒 ≔ 𝑓 − 1 other events • 𝑓𝑞 𝑒 + 1 = 1 ⇒ ℙ ∩ 𝑗 𝐹 𝑗 > 0 ⇒ formula is satisfiable! ∎

Recalling Ramsey Theorem 1.5.2 (Erd ő s, 1947) As 𝑙 → ∞ , we have 1 𝑙 . 𝑆 𝑙 ≥ + 𝑝 1 𝑙 2 𝑓 2 • Disjoint sets of edges are independent • Can improve bound with the local lemma Theorem 4.1.3 As 𝑙 → ∞ , we have 2 𝑙 . 𝑆 𝑙 ≥ 𝑓 + 𝑝 1 𝑙 2

Setting Up the Proof Events 1 • We take 𝐻 ∼ 𝐻 𝑜, 2 as before 𝑜 𝑑 • For 𝐽 ∈ 𝑙 , event 𝐹 𝐽 = 𝐻 𝐽 ≅ 𝐿 𝑙 or 𝐿 𝑙 • ℙ 𝐹 𝐽 = 2 1− 𝑙 2 for all 𝐽 Dependencies • 𝐹 𝐽 independent of all events with disjoint edge-sets • ⇒ 𝐹 𝐽 depends on at most 𝑙 𝑜−2 k−2 − 1 other events 2 𝑙 𝑜−2 • ⇒ 𝑒 + 1 ≤ 2 k−2 Lovász Local Lemma • ⇒ suffices to show 𝑓2 1− 𝑙 𝑙 𝑜−2 𝑙−2 ≤ 1 2 2

Running the Calculations Estimates 𝑙 2 𝑙 • 2 ≤ 2 𝑙 𝑙 2 𝑙 2 𝑜𝑓 𝑜−2 𝑜 • 𝑙−2 ≤ 𝑙 ≤ 𝑜 2 𝑜 2 𝑙 Bounding 𝑜 𝑓2 − 𝑙 𝑙 𝑙 • ⇒ 𝑓2 1− 𝑙 2 𝑙 4 𝑓𝑙 4 𝑜𝑓 𝑜𝑓 2 𝑙 𝑜−2 𝑙−2 ≤ = 2 𝑙 2 𝑜 2 𝑜 2 𝑙 𝑙 2 𝑙 , parenthetical term is 1 1 • If 𝑜 = 𝑓 2 𝑙 2 • Leading coefficient is then 𝑓 2 𝑙 4 2 1−𝑙 • ⇒ can afford for the parenthetical term to be 2 + 𝑝(1) 𝑙 2 • ⇒ can take 𝑜 = 𝑓 + 𝑝 1 𝑙 2 ∎

Any questions?

§2 The Ramsey Number 𝑆 3, 𝑙 Chapter 4: The Lovász Local Lemma The Probabilistic Method

Returning to 𝑆 3, 𝑙 Corollary 2.1.3 As 𝑙 → ∞ , we have 3 𝑙 2 = 𝑆 3, 𝑙 = 𝑃 𝑙 2 . Ω ln 𝑙 Lower bound • Proven using 𝐻 ∼ 𝐻(𝑜, 𝑞) and alterations Limited dependence • Again, disjoint sets of edges are independent • What does the Local Lemma give?

Analysing the Events Two classes of events 𝑜 • For 𝐽 ∈ 3 , let 𝐹 𝐽 = 𝐻 𝐽 ≅ 𝐿 3 𝑜 𝑑 • For 𝐾 ∈ 𝑙 , let 𝐺 𝐾 = 𝐻 𝐾 ≅ 𝐿 𝑙 Probabilities 𝑜 3 , 𝑞 1 ≔ ℙ 𝐹 𝐽 = 𝑞 3 • For each 𝐽 ∈ 𝑙 2 ≈ 𝑓 −𝑞 𝑙 𝑜 • For each 𝐾 ∈ 𝑙 , 𝑞 2 ≔ ℙ 𝐺 𝐾 = 1 − 𝑞 2 • ⇒ in Lovász Local Lemma, should take 𝑞′ = max {𝑞 1 , 𝑞 2 } • ⇒ optimal to have 𝑞 1 = 𝑞 2 12 ln 𝑙 • ⇒ 𝑞 ≈ 𝑙 2

Analysing the Events Further Edge involvements • Each edge appears in 𝑜 − 2 events 𝐹 𝐽 and 𝑜−2 k−2 events 𝐺 𝐾 Dependencies 𝑜−2 • ⇒ each 𝐹 𝐽 depends on fewer than 𝑒 1 ≔ 3 𝑜 − 2 + other events 𝑙−2 𝑙 n−2 • ⇒ each 𝐺 𝐾 depends on fewer than 𝑒 2 ≔ n − 2 + events 2 k−2 • ⇒ need to take d = max 𝑒 1 , 𝑒 2 = 𝑒 2 in the Local Lemma Bounding 𝑜 3 • Thus 𝑓𝑞 ′ 𝑒 + 1 ≤ 𝑓 𝑞 ′ 3 𝑒 ∼ 𝑓 12 ln 𝑙 𝑙 𝑜−2 𝑜 − 2 + 𝑙 2 2 k−2 𝑙 12 3 𝑓 ln 3 𝑙 12 3 𝑓 ln 3 𝑙 12 3 𝑓 ln 3 𝑙 𝑜𝑓 𝑜−2 𝑜 ≤ 𝑙−2 ≤ 𝑙 ≤ 𝑙 4 𝑜 2 𝑙 2 𝑜 2 𝑙 2 𝑙 • For this to be less than 1 , need 𝑜 = 𝑃 𝑙

Post Mortem Different kinds of events • Triangle events 𝐹 𝐽 : • Probability 𝑞 1 = 𝑞 3 • Depend on relatively few other events • Independent set events 𝐺 𝐾 : 𝑙 • Probability 𝑞 2 = 1 − 𝑞 2 • Depend on many other events A possible remedy • Wasteful to use same probability, dependency bounds for all events • Triangle events are “more independent” • Could afford to let them occur with higher probability • Ideally – track each event’s individual probability and dependencies

Tracking Dependencies Representing dependence • Keep track of dependencies using a directed graph • Events are independent of their non-neighbours Definition 4.2.1 (Dependency digraph) Given events 𝐹 1 , 𝐹 2 , … , 𝐹 𝑛 , a directed graph 𝐸 on the vertices 𝑛 is a dependency digraph if, for each 𝑗 ∈ 𝑛 , the event 𝐹 𝑗 is mutually independent of the set of events 𝐹 𝑘 : 𝑗, 𝑘 ∉ 𝐸 . Why a digraph? • In most applications, digraph will be symmetric • 𝑗, 𝑘 ∈ 𝐸 ⇔ 𝑘, 𝑗 ∈ 𝐸 • Can sometimes help to have flexibility

The Lovász Local Lemma Theorem 4.2.2 (Lovász Local Lemma; Erd ős -Lovász, 1975) Let 𝐹 1 , 𝐹 2 , … , 𝐹 𝑛 be events with a dependency digraph 𝐸 . If there are 𝑦 𝑗 ∈ 0,1 such that ℙ 𝐹 𝑗 ≤ 𝑦 𝑗 ς 𝑗,𝑘 ∈𝐸 1 − 𝑦 𝑘 for all 𝑗 ∈ [𝑛] , then 𝑑 ≥ ς 𝑗 1 − 𝑦 𝑗 . ℙ ∩ 𝑗 𝐹 𝑗 Special case: independent events • Can take 𝐸 to be edge-less • ⇒ suffices to have 𝑦 𝑗 = ℙ 𝐹 𝑗 , and done General case • Dependencies → correction factor ς 𝑗,𝑘 1 − 𝑦 𝑘 • The more dependencies, the smaller this factor • ⇒ need probability of these events to shrink