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Complexity Theory Lecture 7 Lecturer: Moni Naor

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Recap Last week: Non-Uniform Complexity Classes Polynomial Time Hierarchy –BPP in hierarchy –If NP µ P/Poly the hierarchy collapses –Approxiamte Counting in the hierarchy This Week: –Randomized Reductions –#P Completeness of Permanent

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The classes we discussed PSPACE : Complete problems: TQBF, 2-person games, generalized geography, … 3 rd level 3 rd level : Complete problems: VC dimension for succinct sets Approximate #P Containment: Approximate #P… 2 nd level 2 nd level : Complete problems: Min DNF, Succinct Set Cover Containment: Min Circuit, BPP… 1 st level 1 st level : Complete problems: SAT, UNSAT,…lots… Containment: factoring, graph isomorphism… P NP coNP Σ3PΣ3P Π3PΠ3P Δ3PΔ3P PSPACE EXP PH Σ2PΣ2P Π2PΠ2P Δ2PΔ2P #P BPP

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Approximate Counting is in the Hierarchy Theorem : For any f 2 #P there is a g 2 Δ 3 P such that on input x and g(x, ) outputs a (1+ ) approximation to f(x) (1- )g(x, ) · f(x) · (1+ )g(x, ) Proof : key point – method for proving that sets are of certain size The set in question: A = Accept(M(x)) µ {0,1} m #P A function f is in #P if there exists a NTM M running in polynomial time where M(x) has f(x) accepting paths for all x 2 {0,1} n

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Methods for proving sizes of sets Let H be family of hash functions where for h 2 H h:{0,1} m {0,1} ℓ if h 2 H is 1-1 on A then clearly |A| · 2 ℓ –a bit difficult to expect full uniqueness Relaxed property: if there are h 1, h 2, … h ℓ 2 H are such that – for all x 2 A there is an 1 · i · l where for all x’ 2 A\{x} we have h i (x) h i (x’) then we know |A| < ℓ 2 ℓ Claim : for a pair-wise independent family H, if |A| · 2 ℓ-1, then there exist h 1, h 2, … h ℓ 2 H with the no collisions property. Denote with U H (A,ℓ). Claim : for A=Accept(M) we have U H (A,ℓ) 2 Σ 2 P Can eliminate the quantifier on i by explicitly going over all possibilities! Conclusion: with oracles calls to U H (A,ℓ) can find smallest ℓ for which U H (A,ℓ) holds Yields a 2ℓ approximation for f(x) Can use amplification of f(x) by running M k times (from accepting positions). New machine M will have f k (x) by accepting paths No collisions property

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Pair-wise Independent Hash Functions A family of hash functions H={h|h:{0,1} m {0,1} ℓ } is pair-wise independent if: for all x ≠ y, for random h 2 R H the random variables h(x), h(y) are uniformly distributed and independent In particular: Pr[h(x)=h(y)]= 1/2 ℓ Already saw the approximate version for distributed equality testing Pr[h(x)=h(y)] · Constructing H: Construction based on matrix multiplication. –Treat x as a m £ 1 vector. –The function is defined by a random m £ ℓ matrix over GF[2]. H={h A |A 2 GF[2] m £ ℓ and h A (x)=Ax } Nice property: local computation

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Proof of No Collisions Claim For any x 2 A and any x’ 2 A\{x} if h is chosen at random from H Pr h [h(x)=h(x’)]= 1/2 ℓ By the Union bound Pr h [h(x) is unique] ¸ |A| ¢ 1/2 ℓ ¸ 1/2 Hitting set : Construct h 1, h 2, … h ℓ one by one, each time covering at least ½ of elements in A that have not been unique so far –Let A’ be the set not covered so far –For any x 2 A’ and any x’ 2 A\{x} for random h 2 R H Pr h [h(x) is unique] ¸ |A| ¢ 1/2 ℓ ¸ ½. Note: this is the full A

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Checking No Collisions Claims Given h 1, h 2, … h ℓ and A (in the form of Accept(M) ) how to check the no collisions claim: First observation: 8 x 2 A 9 i 2 {1,…, ℓ } 8 x’ 2 A\{x}: h i (x) h i (x’) this is in 3 P Quantification over i ‘s is limited: can change order and replace with explicit enumeration 8 x,x 1,x 2,…,x ℓ 2 A if x {x 1,x 2,…,x ℓ } then h 1 (x) h 1 (x 1 ) Ç h 2 (x) h 2 (x 2 ) Ç … Ç h ℓ (x) h ℓ (x ℓ ) this is in 1 P=CoNP

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Second Proof of No Collisions Claim For any x 2 A and any x’ 2 A\{x} if h is chosen at random from H Pr h [h(x)=h(x’)]= 1/2 ℓ By the Union bound Pr h [h(x) is unique] ¸ |A| ¢ 1/2 ℓ ¸ ½. Choose h 1, h 2, … h ℓ at once. –For every x 2 A for each i Pr[h i (x) is unique] ¸ ½. –Probability that at least one i is unique for x is at least 1-1/2 ℓ –With probability at least ½ choice is good for all x 2 A. Conclusions: –There is a choice that is good for all x 2 A. –Approximate Counting is in BPP NP –Choose h 1, h 2, … h ℓ and check the no collisions claim

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Approximate Counting and Uniform Generation For self reducible problems: – Exact counting implies exact uniform generation of witnesses/solutions Choose to set x 1 =1 with probability proportional to # (1,x 2,…x n )/# (x 1,x 2,…x n ) –Approximate counting: almost uniform generation of solutions Accumulated error is multiplicative (1+p(n)) Theorem : if P=NP then given Boolean circuit C possible to sample almost uniformly at random satisfying assignment to C Reverse is also true: from uniform sampling to approximate counting

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From Uniform Generation to Approximate Counting Reverse is also true: from uniform sampling to approximate counting For #SAT we know # (x 1,x 2,…x n ) = # (1,x 2,…x n ) + # (0,x 2,…x n ) To get an approximation: generate n 2 / 2 assignments Learn the ration # (1,x 2,…x n ) and # (0,x 2,…x n ) Conclusion : Uniform Generation to Approximate Counting are the same for self reducible problems Spun a whole industry of Markov Chain/Monte Carlo algorithms Notable successes: Approximating the volume of a convex body Approximating the permanent

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More on BPP in the Hierarchy Simultaneous Provers BPP ½ 2 PTo prove BPP ½ 2 P we considered a game between and players – player wants to prove x L and player wants to prove x L 2 PIn 2 P setting first makes a move and then player (based on ’s move) What if both players have to move simultaneousl y? Want strategy where – player wins whenever x L even if player makes move after player – player wins whenever x L even if player makes move after player Homework: Phrase the above requirements in terms of a polynomial time relation R(x,y,z) BPPShow that for all L BPP such a strategy exists

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Homework problem discussion: low memory device A low memory device is given a stream of a 1, a 2 … a n and then b 1, b 2 … b n where a i, b i 2 {0,1} ℓ and has to determine whether the two multisets are equal. whether there is a a permutation of the a ' s that gives the b's Solution: fix a finite field F (larger than 2 ℓ ) and choose random z 2 F Compute (as you go) i=1 n (z-a i ) and i=1 n (z-b i ) –if the two values are not equal – then two streams are not equal –if the two values are equal – then two streams are equal –except with probability at most n/|F| Can get slightly better probability using -bias probability spaces Degree n poly in z

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Application: low memory verification of satisfiability Suppose that there is a 3-CNF a 3-CNF and a low memory device wishes to verify that it is satisfiable. Device has access to : can verify whether the j th clause of is a given one Prover : To convince the device that a given assignment satisfies : 1.Send first in order of clauses for each clause (x 1 Ç x 2 Ç x 3 ) the value of the assignment on x 1, x 2, x 3 2.Send in order of variables for each variable x i : (number of times appears in , value of the assignment on x i ) Verifier : verify that clauses are correct and assignment satisfies each clause. Problem: how to check consistency between occurrences of x i translate into two streams of a i and b i –each occurrence of x i in first part to a i =(i, x i ) –each occurrence in second part to b i =(i, x i ) (repeat for number of occurrences in ) Check equality of streams

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Problems with unique solutions An instance of SAT may have many satisfying assignments Does the hardness of SAT stem from the multiple assignments –Maybe it is harder to walk towards the right solution if there are several of them Promise problem: we are told that the number of satisfying assignments is 0 or 1. Can we determine which is the case? –Not responsible if more than a single assignment

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Randomized reduction to unique solutions Valiant-Vazirani Theorem : there is a randomized poly-time procedure that given a 3- CNF formula ( x 1, x 2, …, x n ) outputs a 3-CNF formula ’ such that: –if is not satisfiable, then ’ is not satisfiable –if is satisfiable, then with probability at least 1/(8n): ’ has exactly one satisfying assignment Idea of procedure: project the assignments via hashing and look for the event “ only one satisfying assignment hashed to 0 k” k should be log number of satisfying assumptions want to express it uniquely

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Expressing uniqueness Want to use hashing via matrix multiplication –Need to express the statement (x 1, x 2, …, x n ) (y 1, y 2, …, y n )=0 over GF[2] with a unique satisfying assignment Claim: for each Y=( y 1, y 2, …, y n ) there exists a 3-CNF formula θ Y on x 1, x 2, …, x n and additional variables such that: –| θ Y | = O(n) –θ Y is satisfiable iff an even number of variables in {x i y i } i=1 n are 1 –for each setting of the x i variables, this satisfying assignment is unique Proof : scan from left to right computing the inner product so far Introduce z 0, z 1, …, z n and add clause(s) for i=1, …, n –(z i = z i-1 ) if y i =0 –(z i = z i-1 © x i ) if y i =1 and (z 0 =1) and (z n =1) Not surprising: can do for any function in P using circuit simulation

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The Reduction Procedure: –Choose random k 2 R {0,1,2,…,n-1} –for i = 1, 2, …, k pick random subset Y i 2 R {0,1}^n –output ’ = k = θ Y 1 θ Y 2 θ Y k Let T be the set of satisfying assignments for Claim : if |T| > 0, then Pr k {0,1,2,…,n-1} [2 k-2 ≤ |T| ≤ 2 k-1 ] ≥ 1/n Good guess

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Unique Solutions Denote (Y 1 X, Y 2 X, Y K X) with h(X) Claim : if 2 k-2 ≤ |T| ≤ 2 k-1 (good guess), then the probability k has exactly one satisfying assignment is at least 1/8 Fix X T. Let event A x be “ h(X)=0 and for all X’ T\{X}: h(X’) ≠ 0” –Pr[h(X)= 0] = (½) k. –For all X’ T\{X}: Pr[h(X’) = 0| h(X) = 0] = (½) k –Pr[ 8 X’ T\{X} h(X’) ≠ 0| h(X)] ¸ 1-|T|/2 k ¸ ½. –Pr[A k ] ¸ (½) k+1 Since events A x are disjoint from each other Pr[ k has exactly one satisfying assignment] ¸ |T| (½) k+1 ¸ 1/8 At most 2 k-1 At least 2 k-2

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No RP Algorithms for Unique SAT unless… Since the good guess event is independent of the choice of Y 1, Y 2, …, Y k we get the probability of success is at least 1/(8n). RP NP=RP Corollary: if there is an RP algorithm for deciding Unique SAT, then NP=RP Proof: If there is a an algorithm for deciding Unique SAT then there one for finding the satisfiable assignment if it is unique Self reducibility Run the reduction several times and then run the unique sat finding algorithm Can make no errors if not satisfiable

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Alternative Proof Isolation Lemma Let (X,S) be a set system –Each s 2 S is a subset of ground set X –|X|=m Let w: X {1,…, 2m} be a positive weight function chosen uniformly and independently at random in {1,…, 2m} Then Pr[ set with minimum weight is unique in S] ¸ ½.

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Proof of Isolation Lemma For each element i 2 X and weights for all elements in X\{i} w 1, w 2, …, w i-1, w i+1 …, w m can consider critical value t 2 Z for w(i): t may be negative or larger than 2m If w(i) < t, then minimum weight set must contain i If w(i) > t, then minimum weight set does not contain i Pr[w(i)=t] · 1/2m Claim: If no i 2 X is at critical value (w.r.t. other weight), then minimum is unique Proof: If two different sets s 1 and s 2 are the minimum weight sets then there is and i 2 s 1 and i s 2 (or vice versa). Pr[some w(i) is at critical value] · m ¢ 1/2m · ½.

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Applications of the Isolation Lemma Minimum weight matching : given a graph G=(V,E) with m edges assign random weights to edges from {1,…, 2m}. With probability at least ½ the minimum weight perfect matching is unique –Let X=E and S={U ½ E| U is a perfect matching } and apply the Isolation Lemma Homework : use the Isolation Lemma to show another reduction to unique sat You may find it most useful to consider the max clique problem

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Reductions between functions Reduction from function problem A to function problem B –two efficiently computable functions Reduction and ReconstructionIn functions, reducing one problem A to another B involves 1.A mapping of the instances x of A to instances y of B 2.A reconstruction process: answering A(x) from the answer to B(y) xy B(y) A(x) Reduction BA Reconstruction

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Reductions and #P #PA function f is in #P if there exists a NTM M running in polynomial time where M(x) has f(x) accepting paths for all x 2 {0,1} n problem f is #P -complete if –f is in #P FP f –every problem in #P is in FP f Parsimonious Reductions: For counting problems we call a reduction parsimonious if the reconstruction is the identity - preserves the number of solutions –many standard NP -completeness reductions are parsimonious –therefore: if #SAT is #P -complete we get lots of #P -complete problems par·si·mo·ni·ous adj. Excessively sparing or frugal. More expressive term: witness preserving reductions

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#SAT #SAT: given a 3-CNF how many satisfying assignments to are there? Theorem : #SAT is #P -complete. Proof : –clearly in #P: ( ,x) R x satisfies –take any f #P defined by relation R P Essentially Cook-Levin Reduction, but let’s look inside

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#P-Completeness of #SAT –Add new variables z, produce #-CNF such that z (x, y, z) = 1 C(x, y) = 1 –For (x, y) such that C(x, y) = 1 this z is unique Can express for each gate using 3-CNF that z i is the correct computation –hardwire x –# satisfying assignments = |{y : (x, y) R}| …x… …y… C Circuit for R 1 iff (x, y) R f(x) =|{y : (x, y) R}| Homework : show that #Clique is #P-Complete

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Relationship to other classes To compare to classes of decision problems, usually consider P #P Easy connections: NP, coNP P #P P #P PSPACE Toda’s Theorem : PH P #P. So approximate #P PH P #P Big surprise: counting can express iterating quantifiers

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Counting vs. Deciding Question: are problem #P complete because they require finding NP witnesses? –or is the counting difficult by itself? Already saw one counter-example: #DNF But not really convincing given tight connection to CNF

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Bipartite Matching Definition : Let G = (U, V, E) be a bipartite graph with |U| = |V| –a perfect matching in G is a subset M E that touches every node exactly once Long history as motivating problem in Complexity Theory Edmonds, Edmonds-Karp

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Bipartite Matchings #MATCHING : given a bipartite graph G = (U, V, E) how many perfect matchings does it have? Theorem (Valiant): #MATCHING is #P-complete. But: can find a perfect matching in polynomial time! No seemingly related problem that is NP-Complete –counting itself must be difficult The fact that computing #MATCHING is NP-Hard is surprising in itself. The #P completeness is an extra bonus

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The Permanent The permanent of a matrix A is defined as: per(A) = Σ π Π i A i, π(i) Compare to the determinant of a matrix A det(A) = Σ π sgn(π) Π i A i, π(i) # of perfect matchings in a bipartite graph G is exactly the permanent of G ’s adjacency matrix A G –a perfect matching defines a permutation that contributes 1 to the sum

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The Permanent Valiant’s Theorem says that the permanent of {0,1} matrices is #P- complete –permanent has many nice properties that make it a favorite of complexity theory Contrast permanent (hard) per(A) = Σ π Π i A i, π(i) to determinant (easy): det(A) = Σ π sgn(π) Π i A i, π(i) As a result computing per(A) mod 2 is easy Important difference between the two: –det(A B) = det(A) det(B) for all matrices –but per(A B) ≠ per(A) per(B) in general Homework: show

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Cycle Covers Bipartite graph G = (U, V, E) –Adjacency square matrix A G –Corresponding directed graph with self loops Cycle cover in a directed graph: collection of node disjoint cycles that covers all the nodes One-to-one correspondence between matchings and cycle covers Can think of representation of permutation in two ways –Explicit enumeration –Cycle decomposition Number of perfect matching = Number of cycle covers in corresponding graph Weighted graph – the permanent of general matrices –Sum over all cycle covers –For each cycle cover take the product of the weights –Negative weights can zero out positive ones Green edge: weight -1 Black edge: weight 1 Cycle cover 0 If this gadget appears as a subgraph – zeroes out the whole cycle cover

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Proof Structure Show the hardness of the permanent problem for general integer matrices –Build a series of gadgets (weighted graphs) –For a given a 3CNF formula with n variables and m clauses map each satisfying assignment into cycle covers of weight 4 m of weight non-satisfying assignments yield cycle covers whose total contribution is 0 Reduce the computation to {0,1} matrices

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The Clause Gadget All edge weights are all 1 Claim: All cycle covers exclude at least one external edge –Any excluded subset of external edges corresponds to one cycle cover The inner node will not be connected to any other node Outer nodes will be connected

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Literal Gadget All edge weights are 1 Two possible cycle covers: – Red : the literal is false – Black: the literal is true One edge per clause … The gray nodes are the only ones connected to other parts

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Xor Gadget Edge weight not specified are 1 Claim: – Exactly one of {(u,u’), (v,v’)} is used for non-zero contribution to the cycle covers –If Exactly one of {(u,u’), (v,v’)} is used the contribution is 4 Need to check permanent of 4 submatrices 3 u 2 u’ v v’ u,u’,v,v’ appear elsewhere in the graph. Both (u,u’) and (v,v’) are edges.

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Embedding the Gadgets For each clause make a clause gadget –Identify each external edge with a literal Connect each external edge with its literal via an Xor gadget Connect the two literals of variable via an Xor gadget Claim : total weight of cycle covers is s 4 m where s is the number of satisfying assignments Any assignment that does comply with the restrictions adds zero

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Simulating integer weights For small positive weights –Weight 3 For negative weights consider the permanent mod N=2 ℓ +1 where N ¸ 4 m s –The -1 becomes N-1 = 2 ℓ

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Simulating negative/large weights Simulating edge weight 2 ℓ … … ℓ such structures The integer permanent now allows computing mod N permanent which is equal to 4 m ¢ # the satisfying assignments

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References Unique Sat : Leslie Valiant and Vijay Vazirani, 1985 Isolation Lemma : Ketan Mulmuley, Umesh Vazirani and Vijay Vazirani –Motivation: parallel randomized algorithm for matching based on matrix inversion. Checking with low memory devices: –Blum and Kannan, Lipton #P-Completeness of the Permanent : Valiant, 1979.

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