New Constructive Aspects of the Lov´ asz Local Lemma, and their Applications Aravind Srinivasan University of Maryland, College Park June 15, 2011 Collaborators: Bernhard Haeupler (MIT) & Barna Saha (UMD) Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
Algorithmic versions of the LLL A = { A 1 , A 2 , . . . , A m } : “bad” events, each defined by indep. random variables X 1 , X 2 , . . . , X n . Ubiquitous version with neigborhood relation Γ on A . Are all A i simultaneously avoidable? Output = assignment to all X j ; output size = n . Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
Algorithmic versions of the LLL A = { A 1 , A 2 , . . . , A m } : “bad” events, each defined by indep. random variables X 1 , X 2 , . . . , X n . Ubiquitous version with neigborhood relation Γ on A . Are all A i simultaneously avoidable? Output = assignment to all X j ; output size = n . Main results: “Any” LLL application → poly( n )-time alg. (even if m ≫ poly( n )), if we give a tiny slack in the LLL-condition; MAX SAT–like problems: avoiding “most” A i (algorithmically) – interpolation between linearity of expectation and LLL. Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL: symmetric version “Pr[no A i ] > 0”: Union Bound � i Pr[ A i ] < 1 often too weak. LLL (symmetric version): Suppose max i Pr[ A i ] ≤ p , and each A i has ≤ D neighbors. Then, e · p · ( D + 1) ≤ 1 implies Pr[no A i holds] > 0. Numerous applications. Typical case: D ≪ m . Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL: symmetric version “Pr[no A i ] > 0”: Union Bound � i Pr[ A i ] < 1 often too weak. LLL (symmetric version): Suppose max i Pr[ A i ] ≤ p , and each A i has ≤ D neighbors. Then, e · p · ( D + 1) ≤ 1 implies Pr[no A i holds] > 0. Numerous applications. Typical case: D ≪ m . Algorithmic version? Pr[ � i A i ] inevitably small: Choose indep. set I of the A i with | I | ≥ m / ( D + 1). i ∈ I A i ] = (1 − p ) m / ( D +1) ≈ exp( − mp / D ). Pr[ � i A i ] ≤ Pr[ � Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
Application: Domatic Partitions Feige-Halld´ orsson-Kortsarz-S: a maximization problem with a logarithmic apx. threshold. Graph G ; N + ( v ) = inclusive neighborhood of vertex v . Partition vertices into a max. # dominating sets: i.e., “color” vertices with max. # colors so that ∀ vertices v , all colors visible in N + ( v ) . [Chen-Jamieson-Balakrishnan-Morris]: wireless coordination. If ( δ, ∆) = (min., max.) degrees, OPT ≤ δ + 1. Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
Application: Domatic Partitions Feige-Halld´ orsson-Kortsarz-S: a maximization problem with a logarithmic apx. threshold. Graph G ; N + ( v ) = inclusive neighborhood of vertex v . Partition vertices into a max. # dominating sets: i.e., “color” vertices with max. # colors so that ∀ vertices v , all colors visible in N + ( v ) . [Chen-Jamieson-Balakrishnan-Morris]: wireless coordination. If ( δ, ∆) = (min., max.) degrees, OPT ≤ δ + 1. [FHKS]: apx. threshold = ln ∆. Here: 3 ln d –apx. for d -regular G . Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
Domatic partitions assuming d -regularity Randomly color vertices using ℓ ∼ d / (3 ln d ) colors. Bad event A v , c : “ c not visible at v ”. p = Pr[ A v , c ] = (1 − 1 /ℓ ) d +1 ∼ 1 / d 3 . Dependence of fixed A v , c ? Only on A w , c ′ with dist( v , w ) ≤ 2. # w < d 2 ; # c ′ ≤ ℓ . So, D < d 3 / (3 ln d ). e · p · ( D + 1) ≤ 1; thus ∃ good coloring. Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
Domatic partitions assuming d -regularity Randomly color vertices using ℓ ∼ d / (3 ln d ) colors. Bad event A v , c : “ c not visible at v ”. p = Pr[ A v , c ] = (1 − 1 /ℓ ) d +1 ∼ 1 / d 3 . Dependence of fixed A v , c ? Only on A w , c ′ with dist( v , w ) ≤ 2. # w < d 2 ; # c ′ ≤ ℓ . So, D < d 3 / (3 ln d ). e · p · ( D + 1) ≤ 1; thus ∃ good coloring. Correct constant ′′ 3 ′′ → “1 ′′ : iterated app. of LLL, a powerful methodology ([Molloy-Reed]: “Graph colouring and the probabilistic method”). Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL: Asymmetric Version LLL, general “asymmetric” version: If ∃ x : A → (0 , 1) such that � ∀ i : Pr[ A i ] ≤ x ( A i ) (1 − x ( A j )) , A j ∈ Γ( A i ) then Pr[ � i A i ] ≥ � i (1 − x ( A i )) > 0. Numerous applications: (Hyper-)Graph Colorings and Ramsey Numbers Routing [Leighton-Maggs-Rao] LP-Integrality gaps [Feige, Leighton-Lu-Rao-S] Edge-disjoint paths [Andrews] ... Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
The Trivial Algorithm The Trivial Algorithm: repeat pick a random assignment for all X j until no A i holds Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
The Trivial Algorithm The Trivial Algorithm: repeat pick a random assignment for all X j until no A i holds Theorem (LLL) If the LLL-conditions hold, then the above algorithm finds a satisfying assignment with positive probability. Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
The Trivial Algorithm The Trivial Algorithm: repeat pick a random assignment for all X j until no A i holds Theorem (LLL) If the LLL-conditions hold, then the above algorithm finds a satisfying assignment with positive probability. BUT: Run-time usually exponential in m (let alone n ). Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
The Moser-Tardos Breakthrough Algorithmic versions of the LLL: [Beck, Alon, Molloy-Reed, Czumaj-Scheideler, S, Moser, ...] culminating in MT: The MT Algorithm: start with an arbitrary assignment while ∃ event A i that holds do assign new random values to the variables of A i Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
The Moser-Tardos Breakthrough Algorithmic versions of the LLL: [Beck, Alon, Molloy-Reed, Czumaj-Scheideler, S, Moser, ...] culminating in MT: The MT Algorithm: start with an arbitrary assignment while ∃ event A i that holds do assign new random values to the variables of A i Theorem (MT) If the LLL-conditions hold, then the above algorithm finds a x ( A i ) satisfying assignment within an expected � 1 − x ( A i ) iterations. i Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL-distribution and the MT-Algorithm The trivial algorithm outputs a random sample from the conditional LLL-distribution D , the distribution that conditions on avoiding all A i . Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL-distribution and the MT-Algorithm The trivial algorithm outputs a random sample from the conditional LLL-distribution D , the distribution that conditions on avoiding all A i . A Well-known Bound For any event B = B ( X 1 , X 2 , . . . , X n ), − 1 � � � � � � Pr D ( B ) := Pr B A i ≤ Pr ( B ) · (1 − x ( A j )) i A j ∈ Γ( B ) (1) Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL-distribution and the MT-Algorithm The trivial algorithm outputs a random sample from the conditional LLL-distribution D , the distribution that conditions on avoiding all A i . A Well-known Bound For any event B = B ( X 1 , X 2 , . . . , X n ), − 1 � � � � � � Pr D ( B ) := Pr B A i ≤ Pr ( B ) · (1 − x ( A j )) i A j ∈ Γ( B ) (1) Theorem The output distribution of the MT-algorithm satisfies (1). Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL-Applications with Exponentially Many Events Examples: Acyclic edge coloring Non-repetitive coloring Santa Claus problem Edge-disjoint paths, . . . Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
LLL-Applications with Exponentially Many Events Examples: Acyclic edge coloring Non-repetitive coloring Santa Claus problem Edge-disjoint paths, . . . Problems with running MT: x ( A i ) 1 E[# resamplings]: � i 1 − x ( A i ) Aravind Srinivasan University of Maryland, College Park New Constructive Aspects of the Lov´ asz Local Lemma, and their
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