n Boolean variables: x 1, x 2,...,x n 2 {true,false} conjunctive normal form:

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1 Advanced Algorithms

2 k-sat n Boolean variables: x 1, x 2,...,x n 2 {true,false} conjunctive normal form: k-cnf = C 1 ^C 2 ^ ^C m Is φ satisfiable? m clauses: C 1,C 2,...,C m each clause C i = `i1 _ `i2 _ _ `ik is a disjunction of exact k literals each literal: `j 2 {x r, x r } for some r degree d : each clause shares variables with at most d other clauses

3 φ : k-cnf of max degree d Theorem d apple 2 k 2 φ is always satisfiable uniform random assignment for clause Ci, bad event Ai : X 1,X 2,...,X n Ci is not satisfied d apple 2 k 2 Pr n i=1 A i > 0

4 Lovász Sieve Bad events: A 1, A 2,...,A n None of the bad events occurs: Pr n i=1 A i The probabilistic method: being good is possible Pr n i=1 A i > 0

5 events: A1, A2,..., An dependency graph: D(V,E) V = { 1, 2,..., n } ij E Ai and Aj are dependent d : max degree of dependency graph A 1 A 2 A 3 A 5 A 4 A 1 (X 1, X 4 ) A 4 (X 4 ) A 2 (X 1, X 2 ) A 5 (X 3 ) A 3 (X 2, X 3 ) X 1,...,X 4 mutually independent

6 events: A1, A2,..., An d : max degree of dependency graph Lovász Local Lemma i, Pr[Ai] p ep(d + 1) 1 Pr n i=1 A i > 0 General Lovász Local Lemma 9x 1,...,x n 2 [0, 1) " n^ Pr 8i, Pr[A i ] apple x i (1 x j ) j i i=1 A i # n (1 x i ) i=1

7 LLL for k-sat φ : k-cnf of max degree d Theorem d apple 2 k 2 satisfying assignment for φ uniform random assignment for clause Ci, bad event Ai : X 1,X 2,...,X n Ci is not satisfied LLL: e(d + 1) apple 2 k Pr n i=1 A i > 0

8 Algorithmic LLL φ : k-cnf of max degree d with m clauses on n variables Theorem d apple 2 k 2 satisfying assignment for φ Theorem (Moser, 2009) satisfying assignment can be d<2 k 3 found in O(n + km logm) w.h.p.

9 φ : k-cnf of max degree d with m clauses on n variables Solve(φ) pick a random assignment x1, x2,..., xn; while unsatisfied clause C Fix(C); Fix(C) replace variables in C with random values; while unsatisfied clause D overlapping with C Fix(D);

10 φ : k-cnf of max degree d with m clauses on n variables Solve(φ) Pick a random assignment x1, x2,..., xn; while unsatisfied clause C Fix(C); Fix(C) replace variables in C with random values; while unsatisfied clause D overlapping with C Fix(D); at top-level: Observation: A clause C is satisfied and will keep satisfied once it has been fixed. # of top-level calls to Fix(C) : m (# of clauses) total # of calls to Fix(C) (including recursive calls): t

11 φ : k-cnf of max degree d with m clauses on n variables Solve(φ) Pick a random assignment x1, x2,..., xn; while unsatisfied clause C Fix(C); Fix(C) replace variables in C with random values; while unsatisfied clause D overlapping with C Fix(D); m recursion trees total # nodes: t total # of random bits: n+tk (assigned bits) Observation: Fix(C) is called assignment of C is uniquely determined

12 m recursion trees total # nodes: t total # of random bits: n+tk (assigned bits) the sequence of random bits can be recovered from: final assignment: + recursion trees: n bits apple mdlog 2 me + t(log 2 d + O(1)) bits for each recursion tree: root: dlog 2 me bits each internal node: apple log 2 d+ O(1) bits

13 m recursion trees total # nodes: t total # of random bits: n+tk (assigned bits) the sequence of random bits is encoded to : apple n + mdlog 2 me + t(log 2 d + 3) bits Incompressibility Theorem (Kolmogorov) N uniform random bits cannot be encoded to substantially less than N bits.

14 m recursion trees total # nodes: t total # of random bits: n+tk (assigned bits) the sequence of random bits is encoded to : apple n + mdlog 2 me + t(log 2 d + 3) bits Incompressibility Theorem (Kolmogorov) N uniform random bits cannot be encoded to less than N - l bits with probability 1-O(2 -l ).

15 m recursion trees total # nodes: t total # of random bits: n+tk (assigned bits) the sequence of random bits is encoded to : apple n + mdlog 2 me + t(log 2 d + c) bits t(k c log 2 d) apple mdlog 2 me + log n whp when d<2 k c t apple mdlog 2 me + log n k c log 2 d total running time: n+tk = O(n + km logm)

16 Algorithmic LLL φ : k-cnf of max degree d with m clauses on n variables = C 1 ^C 2 ^ ^C m Theorem (Moser, 2009) d<2 k c satisfying assignment can be found in O(n + km logm) whp Solve(φ) Pick a random assignment x1, x2,..., xn; while unsatisfied clause C Fix(C); Fix(C) replace variables in C with random values; while unsatisfied clause D overlapping with C Fix(D);

17 events: A1, A2,..., An d : max degree of dependency graph Lovász Local Lemma i, Pr[Ai] p ep(d + 1) 1 Pr n i=1 A i > 0 General Lovász Local Lemma 9x 1,...,x n 2 [0, 1) " n^ Pr 8i, Pr[A i ] apple x i (1 x j ) j i i=1 A i # n (1 x i ) i=1

18 Constraint Satisfaction Problem variables: x1, x2,..., xn D (domain) constraints: C1, C2,..., Cm where C i (x i1,x i2,...) 2 {true, false} CSP solution: an assignment of variables satisfying all constraints examples: SAT, graph colorability,... existence: When does a solution exist? search: How to find a solution?

19 The Probabilistic Method CSP C1, C2,..., Cm defined on x1, x2,..., xn sampling random values of x1, x2,..., xn Bad event Ai: constraint Ci is violated None of the bad events occurs with prob: Pr " m^ i A i # The probabilistic method: being good is possible " m^ Pr i=1 A i # > 0

20 events: A1, A2,..., An d : max degree of dependency graph Lovász Local Lemma i, Pr[Ai] p ep(d + 1) 1 Pr n i=1 A i > 0 General Lovász Local Lemma 9x 1,...,x n 2 [0, 1) " n^ Pr 8i, Pr[A i ] apple x i (1 x j ) j i i=1 A i # n (1 x i ) i=1

21 mutually independent random variables: X X bad events: A A defined on variables in X vbl(a) X: set of variables on which A is defined neighborhood: Γ(A) = { B A B A and vbl(a) vbl(b) } inclusive neighborhood: Γ + (A) = Γ(A) { A } events that are dependent with A, excluding/including A itself Lovász Local Lemma (general) 9 : A! [0, 1) 8A 2 A : Pr[A] apple A (1 B ) B2 (A) " ^ Pr A2A A # > 0 (1 A ) A2A

22 mutually independent random variables: X X bad events: A A defined on variables in X vbl(a) X: set of variables on which A is defined neighborhood: Γ(A) = { B A B A and vbl(a) vbl(b) } inclusive neighborhood: Γ + (A) = Γ(A) { A } events that are dependent with A, excluding/including A itself Lovász Local Lemma (general) 9 : A! [0, 1) 8A 2 A : Pr[A] apple A B2 (A) (1 B ) values of variables in X avoiding all bad events A A simultaneously.

Algorithmic LLL bad events A A defined on mutually independent random variables X X vbl(a): set of variables on which A is defined neighborhood Γ(A) and inclusive neighborhood Γ + (A) Assumption: I.

23 Algorithmic LLL bad events A A defined on mutually independent random variables X X vbl(a): set of variables on which A is defined neighborhood Γ(A) and inclusive neighborhood Γ + (A) Assumption: I. We can efficiently sample an independent evaluation of every random variable X X. II. We can efficiently check the violation of every event A A. RandomSolver: sample all X X; while a non-violated bad event A A: resample all X vbl(a);

24 bad events A A defined on mutually independent random variables X X vbl(a): set of variables on which A is defined neighborhood Γ(A) and inclusive neighborhood Γ + (A) RandomSolver: sample all X X; while a non-violated bad event A A: resample all X vbl(a); Moser-Tardos 2010: 9 : A! [0, 1) 8A 2 A : Pr[A] apple A B2 (A) (1 B ) RandomSolver finds values of all X X avoiding all A A X within expected A 1 resamples. A A2A

25 bad events A A defined on mutually independent random variables X X vbl(a): set of variables on which A is defined neighborhood Γ(A) and inclusive neighborhood Γ + (A) RandomSolver: sample all X X; while a non-violated bad event A A: resample all X vbl(a); Moser-Tardos 2010: A A, Pr[A] p ep(d + 1) 1 where d=max A Γ(A) RandomSolver finds values of all X X violating all A A within expected A /d resamples.

26 k-sat φ : k-cnf of max degree d with m clauses on n variables RandomSolver: pick a random assignment x1, x2,..., xn; while an unsatisfied clause C: replace variables in C with random values; d apple 2 k 2 ( e(d+1) 2 k ) RandomSolver returns a satisfying assignment within expected O(n + km/d) time

27 bad events A A defined on mutually independent random variables X X vbl(a): set of variables on which A is defined neighborhood Γ(A) and inclusive neighborhood Γ + (A) RandomSolver: sample all X X; while a non-violated bad event A A: resample all X vbl(a); Moser-Tardos 2010: 9 : A! [0, 1) 8A 2 A : Pr[A] apple A B2 (A) (1 B ) RandomSolver finds values of all X X avoiding all A A X within expected A 1 resamples. A A2A

28 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); execution log Λ: Λ1, Λ2, Λ3,... A random sequence of resampled events NA = { i Λi=A } total # of times that A is resampled Moser-Tardos 2010: 9 : A! [0, 1) 8A 2 A : Pr[A] apple A B2 (A) 8A 2 A : E[N (1 B ) A ] apple A 1 A

29 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); execution log Λ: Λ1, Λ2, Λ3,... A random sequence of resampled events witness tree: A witness tree τ is a labeled tree in which every vertex v is labeled by an event Av A, such that siblings have distinct labels. T(Λ, t) is a witness tree constructed from exe-log Λ: initially, T is a single root with label Λt for i = t-1, t-2,...,1 if a vertex v in T with label Av Γ + (Λi) add a new child u to the deepest such v and label it with Λi T(Λ, t) is the resulting T

30 dependency graph: B A E C D exe-log Λ: D, C, E, D, B, A, C, A, D,... T(Λ, 8): B A A E D C T(Λ, t) is a witness tree constructed from exe-log Λ: initially, T is a single root with label Λt for i = t-1, t-2,...,1 if a vertex v in T with label Av Γ + (Λi) add a new child u to the deepest such v and label it with Λi T(Λ, t) is the resulting T

31 dependency graph: B A E C D exe-log Λ: D, C, E, D, B, A, C, A, D,... T(Λ, 8): D B A C A E T(Λ, 9): T(Λ, t) is a witness tree constructed from exe-log Λ: initially, T is a single root with label Λt for i = t-1, t-2,...,1 if a vertex v in T with label Av Γ + (Λi) add a new child u to the deepest such v and label it with Λi T(Λ, t) is the resulting T B C E D D D C

32 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); execution log Λ: Λ1, Λ2, Λ3,... A random sequence of resampled events witness tree: A witness tree τ is a labeled tree in which every vertex v is labeled by an event Av A, such that siblings have distinct labels. T(Λ, t) is a witness tree constructed from exe-log Λ: initially, T is a single root with label Λt for i = t-1, t-2,...,1 if a vertex v in T with label Av Γ + (Λi) add a new child u to the deepest such v and label it with Λi T(Λ, t) is the resulting T T(Λ, s) T(Λ, t) if s t E[N A ]= X 2T A Pr[9t, T (,t)= ] TA: set of all witness trees with root-label A

33 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); execution log Λ: Λ1, Λ2, Λ3,... A random sequence of resampled events T(Λ, t) is a witness tree constructed from exe-log Λ: initially, T is a single root with label Λt for i = t-1, t-2,...,1 if a vertex v in T with label Av Γ + (Λi) add a new child u to the deepest such v and label it with Λi T(Λ, t) is the resulting T Lemma 1 For any particular witness tree τ: Pr[9t, T (,t)= ] apple v2 Pr[A v ]

34 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); execution log Λ: Λ1, Λ2, Λ3,... A random sequence of resampled events Lemma 1 For any particular witness tree τ: Pr[9t, T (,t)= ] apple v2 Pr[A v ] NA = { i Λi=A } total # of times that A is resampled E[N A ]= X Pr[9t, T (,t)= ] 2T A X (lemma 1) apple Pr[A v ] TA: set of all witness trees with root-label A 2T A v2

35 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); LLL condition: 8A 2 A : (lemma 1) (LLL cond.) Pr[A] apple A E[N A ]= X Pr[9t, T (,t)= ] 2T X A apple Pr[A v ] 2T A apple X 2T A 9 : A! [0, 1) v2 v2 B2 (A) 2 4 (A v ) execution log Λ: random sequence of resampled events (1 B ) B2 (A v ) Λ1, Λ2, Λ3,... A 3 (1 (B)) 5 TA: set of all witness trees with root-label A goal: apple A 1 A

36 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); execution log Λ: Λ1, Λ2, Λ3,... A random sequence of resampled events Lemma 1 For any particular witness tree τ: Pr[9t, T (,t)= ] apple v2 Pr[A v ] X (t) i : t-th sampling of variable Xi X exe-log Λ: D,C,E,D,B,A,C,A,D,... X 1 : X (0) 1,X(1) 1,X(2) 1,X(3) 1,X(4) 1,... B(X 2,X 3 ) A X 2 : X (0) 2,X(1) 2,X(2) 2,X(3) 2,X(4) 2,... X 3 : X (0) 3,X(1) 3,X(2) 3,X(3) 3,X(4) 3,... X 4 : X (0) 4,X(1) 4,X(2) 4,X(3) 4,X(4) 4,... A(X 1,X 2 ) E(X 1,X 4 ) C(X 3,X 4 ) D(X 4 ) D B A C E

37 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); execution log Λ: Λ1, Λ2, Λ3,... A random sequence of resampled events Lemma 1 For any particular witness tree τ: Pr[9t, T (,t)= ] apple v2 Pr[A v ] NA = { i Λi=A } total # of times that A is resampled E[N A ]= X Pr[9t, T (,t)= ] 2T A X (lemma 1) apple Pr[A v ] TA: set of all witness trees with root-label A 2T A v2

38 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); LLL condition: 8A 2 A : (lemma 1) (LLL cond.) Pr[A] apple A E[N A ]= X Pr[9t, T (,t)= ] 2T X A apple Pr[A v ] 2T A apple X 2T A 9 : A! [0, 1) v2 v2 B2 (A) 2 4 (A v ) execution log Λ: random sequence of resampled events (1 B ) B2 (A v ) Λ1, Λ2, Λ3,... A 3 (1 (B)) 5 TA: set of all witness trees with root-label A goal: apple A 1 A

39 grow a random witness tree TA TA : initially, TA is a single root with label A for i = 1, 2,... for every vertex v at depth i (root has depth 1) in TA for every B Γ + (Av): add a new child u to v independently with probability αb; and label it with B; stop if no new child added for an entire level Lemma 2 Pr[T A = ] = 1 For any particular witness tree τ TA: A A 2 4 (A v ) v2 B2 (A v ) 3 (1 B ) 5

40 Lemma 2 Pr[T A = ] = 1 For any particular witness tree τ TA: A A 2 4 (A v ) v2 B2 (A v ) 3 (1 B ) 5 in τ: A + (A) : Pr[T A = ] = 1 A v2 = 1 A A = 1 A A 2 4 (A v ) v2 v2 2 ) + 0 (A) B2 + 0 (A v) 4 (A v) 1 (A v ) 2 4 (A v ) B2 (A v ) 3 (1 B ) 5 B2 + (A v ) (1 B ) 5 3 (1 B ) 5 3

41 RandomSolver: sample all X X; while a non-violated A A: resample all X vbl(a); LLL condition: 8A 2 A : (lemma 1) (LLL cond.) Pr[A] apple A E[N A ]= X Pr[9t, T (,t)= ] 2T X A apple Pr[A v ] 2T A apple X 2T A 9 : A! [0, 1) v2 v2 B2 (A) 2 4 (A v ) execution log Λ: random sequence of resampled events (1 B ) B2 (A v ) Λ1, Λ2, Λ3,... A 3 (1 (B)) 5 (lemma 2) apple A apple X A Pr[T A = ] 1 A 1 A 2T A TA: set of all witness trees with root-label A

bad events A A defined on mutually independent random variables X X vbl(a): set of variables on which A is defined neighborhood Γ(A) and inclusive neighborhood Γ + (A) RandomSolver: sample all X X;

42 bad events A A defined on mutually independent random variables X X vbl(a): set of variables on which A is defined neighborhood Γ(A) and inclusive neighborhood Γ + (A) RandomSolver: sample all X X; while a non-violated bad event A A: resample all X vbl(a); Moser-Tardos 2010: 9 : A! [0, 1) 8A 2 A : Pr[A] apple (A) B2 (A) (1 (B)) RandomSolver finds values of all X X violating all A A X (A) within expected 1 (A) A2A resamples.

Advanced Algorithms 南京大学尹一通

Advanced Algorithms 南京大学尹一通 Advanced Algorithms 南京大学尹一通 Constraint Satisfaction Problem variables: (CSP) X = {x 1, x2,..., xn} each variable ranges over a finite domain Ω an assignment σ ΩX assigns each variable a value in Ω constraints: