Randomness in Computing L ECTURE 23 Last time • Probabilistic method • Lovasz Local Lemma (LLL) • Algorithmic LLL Today • Probabilistic method • Algorithmic LLL • Applications of LLL 4/16/2020 Sofya Raskhodnikova;Randomness in Computing
Algorithmic LLL for 𝒍 SAT Algorithmic Lovasz Local Lemma for 𝑙 SAT If 𝒆 ≤ 𝟑 𝒍−𝟒 = 𝟑 𝒍 𝟗 for some 𝑙 CNF formula 𝜚 , then 𝜚 is satisfiable. Moreover, a satisfying assignment can be found in 𝑃(𝑛 2 log 𝑛) time with probability at least 1 − 2 −𝑛 . Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Moser-Tardos Algorithm for LLL Input: a 𝑙 CNF formula with clauses 𝐷 1 , … , 𝐷 𝑛 on 𝑜 variables and with 𝑒 ≤ 2 𝑙−3 Global variable Let 𝑆 be a random assignment where each variable is 1. assigned 0 or 1 uniformly and independently. While some clause 𝐷 is violated by 𝑆 , run FIX (𝐷) 2. 3. 3. 𝐒𝐟𝐮𝐯𝐬𝐨 𝑆. FIX (𝐷) 1. Pick new values for 𝑙 variables in 𝐷 uniformly and independently and update 𝑆 . While some clause 𝐸 that shares a variable with 𝐷 is 2. violated by 𝑆 , run FIX (𝐸) 𝑬 could be 𝑫 if we chose the same values as before Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Correctness of Moser-Tardos Theorem (Correctness) If Moser-Tardos terminates, it outputs a satisfying assignment. Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Run time of Moser-Tardos • Assume: 𝑛 ≥ 2 𝑙 (o.w. trivial by other means) Theorem (Run time) If 𝒆 ≤ 𝟑 𝒍−𝟒 then Moser-Tardos terminates after 𝑃(𝑛 log 𝑛) resampling steps with probability at least 1 − 2 −𝑛 . • Proof idea: Clever way to ``compress’’ random bits if the algorithm runs for too long. Observation 2 Set A Set B If a function 𝑔: 𝐵 → 𝐶 is injective 𝒈 (i.e., invertible on its range 𝑔(𝐵) ) then 𝐶 ≥ |𝐵| . Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Function 𝒈 𝑼 • Suppose we stop Moser-Tardos after 𝑈 resampling steps. Randomness used: 𝒐 bits for initial assignment 𝒍 bits for each resampling step 𝒐 + 𝑼𝒍 bits Total: • Let 𝐵 be the set of all choices for 𝑜 + 𝑈𝑙 bits 𝑔 𝑈 ( 𝑦 0 , 𝑧 0 = 𝑦 𝑈 , 𝑨 𝑈 initial 𝑼𝒍 bits for assignment transcript assignment reassignment after 𝑼 after 𝑼 resampling resampling steps steps Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Transcript • Each call to FIX gets recorded as follows: If FIX 𝐷 is called by the main algorithm 𝟐 If FIX 𝐸 is a recursive call made by FIX 𝐷 𝟐 • When a call to FIX returns, 𝟏 is written on the transcript Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Run time of Moser-Tardos Lemma 1 Function 𝑔 𝑈 is invertible on all inputs (𝑦 0 , 𝑧 0 ) for which Moser-Tardos does not terminate within 𝑈 steps when run with randomness (𝑦 0 , 𝑧 0 ) . Lemma 2 Length of transcript 𝑨 𝑈 is at most 𝒏(⌈𝐦𝐩𝐡 𝟑 𝒏⌉ + 𝟑) + 𝑼 ⋅ (𝒍 − 𝟐) . Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Proof of Theorem First, consider 𝑈 such that Moser-Tardos never terminates within 𝑈 resampling steps. • There is a valid transcript 𝑨 𝑈 for every choice of the random 𝑜 + 𝑈𝑙 bits needed to run Moser-Tardos Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Proof of Theorem (continued) Now, consider 𝑈 such that Moser-Tardos fails to terminate 1 w.p. ≥ 2 𝑛 within 𝑈 resampling steps. 𝑈 is invertible on the set of size ≥ 2 𝑜+𝑈𝑙−𝑛 • Then 𝑔 Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Proof of Lemma 1 Lemma 1 Function 𝑔 𝑈 is invertible on all inputs (𝑦 0 , 𝑧 0 ) for which Moser-Tardos does not terminate within 𝑈 steps when run with randomness (𝑦 0 , 𝑧 0 ) . Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Algorithmic LLL for 𝒍 SAT Algorithmic Lovasz Local Lemma for 𝑙 SAT If 𝒆 ≤ 𝟑 𝒍−𝟒 = 𝟑 𝒍 𝟗 for some 𝑙 CNF formula 𝜚 , then 𝜚 is satisfiable. Moreover, a satisfying assignment can be found in 𝑃(𝑛 2 log 𝑛) time with probability at least 1 − 2 −𝑛 . Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Lovasz Local Lemma (LLL) • Event 𝐹 is mutually independent from the events 𝐹 1 , … , 𝐹 𝑜 if, for any subset 𝐽 ⊆ [𝑜] , Pr 𝐹 ሩ 𝐹 𝑘 ] = Pr[𝐹] . 𝑘∈𝐽 • A dependency graph for events 𝐶 1 , … , 𝐶 𝑜 is a graph with vertex set [𝑜] and edge set 𝐹 , s.t. ∀𝑗 ∈ 𝑜 , event 𝐶 𝑗 is mutually independent of all events 𝐶 𝑗, 𝑘 ∉ 𝐹} . 𝑘 Lovasz Local Lemma Let 𝐶 1 , … , 𝐶 𝑜 be events over a common sample space s.t. max degree of the dependency graph of 𝐶 1 , … , 𝐶 𝑜 is at most 𝒆 1. ∀𝑗 ∈ 𝑜 , Pr 𝐶 𝑗 ≤ 𝒒 2. If 𝒇𝒒 𝒆 + 𝟐 ≤ 𝟐 then Prځ 𝑗∈ 𝑜 ഥ 𝐶 𝑗 > 0 Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Application of LLL Theorem 𝒍−𝟑 + 1 2 1− 𝒍 𝒍 𝒐 𝟑 ≤ 1 then edges of 𝐿 𝑜 can be colored If 𝒇 𝟑 with 2 colors so that there is no monochromatic 𝐿 𝑙 . Proof: Sofya Raskhodnikova; Randomness in Computing 4/16/2020
Application 2: edge-disjoint paths • 𝒐 pairs of users need to communicate using edge-disjoint paths • ∀𝑗 ∈ 𝑜 , pair 𝑗 can choose a path from collection 𝑄 𝑗 of size 𝒏 . Theorem If ∀𝑗 ≠ 𝑘, each path in 𝑄 𝑗 shares edges with at most 𝒍 paths in 𝑄 𝑘 and 2𝒇𝒐𝒍 ≤ 𝒏 then there is a way to choose 𝒐 edge-disjoint paths. Proof: Sofya Raskhodnikova; Randomness in Computing 4/16/2020
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