Randomness in Computing L ECTURE 23 Last time Probabilistic method - - PowerPoint PPT Presentation

4/16/2020

Randomness in Computing

LECTURE 23

Last time

• Probabilistic method
• Lovasz Local Lemma (LLL)
• Algorithmic LLL

Today

• Probabilistic method
• Algorithmic LLL
• Applications of LLL

Sofya Raskhodnikova;Randomness in Computing

Algorithmic LLL for 𝒍SAT

4/16/2020

Sofya Raskhodnikova; Randomness in Computing

Algorithmic Lovasz Local Lemma for 𝑙SAT

If 𝒆 ≤ 𝟑𝒍−𝟒 = 𝟑𝒍

𝟗 for some 𝑙CNF formula 𝜚, then 𝜚 is satisfiable.

Moreover, a satisfying assignment can be found in 𝑃(𝑛2 log 𝑛) time with probability at least 1 − 2−𝑛.

Moser-Tardos Algorithm for LLL

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Sofya Raskhodnikova; Randomness in Computing

1. Let 𝑆 be a random assignment where each variable is assigned 0 or 1 uniformly and independently. 2. While some clause 𝐷 is violated by 𝑆, run FIX(𝐷) 3. 3. 𝐒𝐟𝐮𝐯𝐬𝐨 𝑆.

Input: a 𝑙CNF formula with clauses 𝐷1, … , 𝐷𝑛

• n 𝑜 variables and with 𝑒 ≤ 2𝑙−3
• 1. Pick new values for 𝑙 variables in 𝐷 uniformly and

independently and update 𝑆. 2. While some clause 𝐸 that shares a variable with 𝐷 is violated by 𝑆, run FIX(𝐸)

FIX(𝐷)

Global variable 𝑬 could be 𝑫 if we chose the same values as before

Correctness of Moser-Tardos

4/16/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem (Correctness)

If Moser-Tardos terminates, it outputs a satisfying assignment.

Run time of Moser-Tardos

• Assume: 𝑛 ≥ 2𝑙 (o.w. trivial by other means)
• Proof idea: Clever way to ``compress’’ random bits

if the algorithm runs for too long.

4/16/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem (Run time)

If 𝒆 ≤ 𝟑𝒍−𝟒 then Moser-Tardos terminates after 𝑃(𝑛 log 𝑛) resampling steps with probability at least 1 − 2−𝑛.

Observation 2

If a function 𝑔: 𝐵 → 𝐶 is injective (i.e., invertible on its range 𝑔(𝐵)) then 𝐶 ≥ |𝐵|.

Set A Set B 𝒈

Function 𝒈𝑼

• Suppose we stop Moser-Tardos after 𝑈 resampling steps.

Randomness used:

• Let 𝐵 be the set of all choices for 𝑜 + 𝑈𝑙 bits

𝑔

𝑈( 𝑦0, 𝑧0 = 𝑦𝑈, 𝑨𝑈

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Sofya Raskhodnikova; Randomness in Computing

𝒐 bits for initial assignment 𝒍 bits for each resampling step 𝒐 + 𝑼𝒍 bits Total:

initial assignment 𝑼𝒍 bits for reassignment assignment after 𝑼 resampling steps transcript after 𝑼 resampling steps

Transcript

• Each call to FIX gets recorded as follows:

If FIX 𝐷 is called by the main algorithm If FIX 𝐸 is a recursive call made by FIX 𝐷

• When a call to FIX returns,

is written on the transcript

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Sofya Raskhodnikova; Randomness in Computing

𝟐 𝟐 𝟏

Run time of Moser-Tardos

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Sofya Raskhodnikova; Randomness in Computing

Lemma 1

Function 𝑔

𝑈 is invertible on all inputs (𝑦0, 𝑧0) for which Moser-Tardos

does not terminate within 𝑈 steps when run with randomness (𝑦0, 𝑧0).

Lemma 2

Length of transcript 𝑨𝑈 is at most 𝒏(⌈𝐦𝐩𝐡𝟑 𝒏⌉ + 𝟑) + 𝑼 ⋅ (𝒍 − 𝟐) .

Proof of Theorem

First, consider 𝑈 such that Moser-Tardos never terminates within 𝑈 resampling steps.

• There is a valid transcript 𝑨𝑈 for every choice of

the random 𝑜 + 𝑈𝑙 bits needed to run Moser-Tardos

4/16/2020

Sofya Raskhodnikova; Randomness in Computing

Proof of Theorem (continued)

Now, consider 𝑈 such that Moser-Tardos fails to terminate w.p. ≥

1 2𝑛 within 𝑈 resampling steps.

• Then 𝑔

𝑈 is invertible on the set of size ≥ 2𝑜+𝑈𝑙−𝑛

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Sofya Raskhodnikova; Randomness in Computing

Proof of Lemma 1

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Sofya Raskhodnikova; Randomness in Computing

Lemma 1

Function 𝑔

𝑈 is invertible on all inputs (𝑦0, 𝑧0) for which Moser-Tardos

does not terminate within 𝑈 steps when run with randomness (𝑦0, 𝑧0).

Algorithmic LLL for 𝒍SAT

4/16/2020

Sofya Raskhodnikova; Randomness in Computing

Algorithmic Lovasz Local Lemma for 𝑙SAT

If 𝒆 ≤ 𝟑𝒍−𝟒 = 𝟑𝒍

𝟗 for some 𝑙CNF formula 𝜚, then 𝜚 is satisfiable.

Moreover, a satisfying assignment can be found in 𝑃(𝑛2 log 𝑛) time with probability at least 1 − 2−𝑛.

Lovasz Local Lemma (LLL)

• Event 𝐹 is mutually independent from the events 𝐹1, … , 𝐹𝑜

if, for any subset 𝐽 ⊆ [𝑜], Pr 𝐹 ሩ

𝑘∈𝐽

𝐹

𝑘] = Pr[𝐹] .

• A dependency graph for events 𝐶1, … , 𝐶𝑜 is a graph with vertex

set [𝑜] and edge set 𝐹, s.t. ∀𝑗 ∈ 𝑜 , event 𝐶𝑗 is mutually independent of all events 𝐶

𝑘

𝑗, 𝑘 ∉ 𝐹}.

4/16/2020

Sofya Raskhodnikova; Randomness in Computing

Lovasz Local Lemma

Let 𝐶1, … , 𝐶𝑜 be events over a common sample space s.t. 1. max degree of the dependency graph of 𝐶1, … , 𝐶𝑜 is at most 𝒆 2. ∀𝑗 ∈ 𝑜 , Pr 𝐶𝑗 ≤ 𝒒 If 𝒇𝒒 𝒆 + 𝟐 ≤ 𝟐 then Prځ𝑗∈ 𝑜 ഥ

𝐶𝑗 > 0

Application of LLL

Proof:

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Sofya Raskhodnikova; Randomness in Computing

Theorem

If 𝒇

𝒍 𝟑 𝒐 𝒍−𝟑 + 1 21− 𝒍

𝟑 ≤ 1 then edges of 𝐿𝑜 can be colored

with 2 colors so that there is no monochromatic 𝐿𝑙.

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 𝑄𝑗 of size 𝒏.

Proof:

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Sofya Raskhodnikova; Randomness in Computing

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.