An IPS for TQBF Intro to Approximability

Transcription

1 An IPS for TQBF Intro to Approximability

2 Outline Proof that TQBF (the language of true (valid) quantified Boolean formulas) is in the class IP Introduction to approximation algorithms for NP optimization problems

3 Recall from last time Simplification Lemma: Any TQBF instance ϕ can be converted in polynomial time into a simple TQBF instance ϕ such that ϕ is valid if and only if ϕ is Arithmetization Lemma: A simple TQBF instance ϕ is valid iff v(aϕ) > 0, where Aϕ is the arithmetization of ϕ

4 Recall from last time Simplification Lemma: Any TQBF instance ϕ can be converted in polynomial time into a simple TQBF instance ϕ such that ϕ is valid if and only if ϕ is Arithmetization Lemma: A simple TQBF instance ϕ is valid iff v(aϕ) > 0, where Aϕ is the arithmetization of ϕ

5 Recall from last time Simplification Lemma: Any TQBF instance ϕ can be converted in polynomial time into a simple TQBF instance ϕ such that ϕ is valid if and only if ϕ is Arithmetization Lemma: A simple TQBF instance ϕ is valid iff v(aϕ) > 0, where Aϕ is the arithmetization of ϕ Computing Mod a Prime Lemma: v(aϕ) > 0 iff there is a prime p in [2 A ϕ, 2 3 A ϕ ] such that v(aϕ) 0 mod p

6 An IPS for TQBF Input: a QBF ϕ; let ϕ have m quantifiers and without loss of generality assume that ϕ is simple Arithmetize ϕ to obtain Aϕ ; let A0 = Aϕ // ϕ is valid iff v(aϕ) > 0 Guess p in the range in [2 A ϕ, 2 3 A ϕ ]; reject if p is not prime Guess a 0 in the range [1,,p-1] // check that v(a0) = a 0 mod p

7 An IPS for TQBF, continued for i from 1 to m do // m is # quantifiers of ϕ 1. let Ai-1 = ci + ci (Ou Ai(u)), where Ou is the leftmost or symbol 2. guess a polynomial αi(u) of degree at most 2 Aϕ 3. check that ci + ci (Ou αi(u)) = ai-1 mod p; if not, reject 4. choose ri randomly and uniformly in the range [0... p-1] 5. let ai = αi(ri) mod p 6. let Ai be the expression Ai(ri) check that v(am) = am mod p; if not, reject and otherwise accept

8 Proof of correctness (outline) All steps of the algorithm run in polynomial time (easy)

9 Proof of correctness (outline) A strategy S(ϕ) is a choice of αi(u) at step 2 of each iteration i of the protocol Claim 1: If v(aϕ) = a 0 mod p then for some strategy S(ϕ), the IPS accepts with probability 1 Claim 2: If v(aϕ) a 0 mod p then for all strategies S(ϕ), the IPS accepts with probability at most 1- (1-2n/2 n ) n Correctness follows, since ϕ is valid iff v(aϕ) 0 mod p, for some prime p in the range specified in the IPS

10 Proof of correctness (outline) Claim 1: If v(aϕ) = a 0 mod p then for some strategy S(ϕ), the IPS accepts with probability 1

11 Proof of correctness (outline) Claim 1: If v(aϕ) = a 0 mod p then for some strategy S(ϕ), the IPS accepts with probability 1 Proof ideas: The strategy S(ϕ) simply returns the polynomial αi(u) that is equal to Ai(u) (mod p) Use the following (proved last time): Low degree lemma: Let ϕ be simple and let Aϕ = Qx Ai (x). Then the degree of polynomial Ai(x) is at most 2 α

12 Proof of correctness (outline) Claim 2: If v(aϕ) a 0 mod p then for all strategies S(ϕ), the IPS accepts with probability at most (1-2n/2 n ) n

13 Proof of correctness (outline) Claim 2: If v(aϕ) a 0 mod p then for all strategies S(ϕ), the IPS accepts with probability at most 1 - (1-2n/2 n ) n Proof ideas: Fix any strategy S = S(ϕ). For each i between 0 and m, let Ei(ϕ,S) be the event that v(ai) ai mod p, or that the protocol rejects before round i+1 is reached (or, if i=m, that the loop terminates) Show by induction that Prob[Ei(ϕ,S)] (1-2n/2 n ) i, where the probability is taken over the choice of ri

14 Summary We ve shown an interactive proof system that accepts TQBF Thus, IP = PSPACE: for any language L in PSPACE a prover can convince a coin-flipping verifier in polynomial time that a yes-instance x is indeed in L, and can fool the verifier with low probability when x is a no-instance of L

15 Summary The IP = PSPACE result raises other questions: If all of PSPACE can be proved (with low error probability) to a computationally limited coinflipping verifier, can we limit the verifier further when proving membership in an NP language with low error probability? We ll come back to this question after a detour to approximation algorithms for NP-hard problems

16 Approximation algorithms

17 Approximation algorithms Motivating example: Max SAT: Given a Boolean formula ϕ in conjunctive normal form, find the maximum number of clauses that can be simultaneously satisfied This is an optimization version of the classical SAT decision problem Exercise: suggest simple algorithms that aim to satisfy as many clauses as possible

18 Approximation algorithms for Max SAT Greedy algorithm: assign a truth value to the variables in turn, choosing a value for variable xi that satisfies at least half of the not-yet-satisfied clauses in which xi appears Even simpler: either the all-true or all-false assignment will satisfies at least half of the clauses (why?)

19 Approximation algorithms for Min Vertex Cover Given an undirected graph G = (V,E), find a minimum vertex cover for G. A vertex cover is a set of nodes that are incident on all edges of G Exercise: suggest simple algorithms that aim to find the smallest possible vertex cover

20 Approximation algorithms for Min Vertex Cover Greedy algorithm: Start with S = Repeat until the graph has no edges: Pick the vertex v that is incident on the most edges (breaking ties arbitrarily), add v to S and remove its incident edges from the graph

21 Approximation algorithms for Min Vertex Cover From The Nature of Computation by Chris Moore

22 Approximation algorithms for Min Vertex Cover Conservative algorithm: Start with S = Repeat until the graph has no edges: Pick any edge e1 of E, and add both its endpoints to S. Delete these two vertices from the graph as well as all edges adjacent to them This algorithm finds a vertex cover of size at most twice the minimum why?

23 Optimization problems An optimization problem has the following properties: Corresponding to an instance I of the problem is a set of solutions SI. Corresponding to each solution δ SI is a value, which is a positive rational number. is either a maximization problem, in which case we want to find the solution with maximum value, or a minimization problem. Let Opt(I) be the value of the optimal solution to I.

24 Optimization problems An algorithm A is an approximation algorithm for if given an instance I of, A computes a solution of I. Let A(I) denote the value of the solution computed by A on instance I. Let RA(I) = max{ A(I)/Opt(I), Opt(I)/A(I) }. Note that 1 RA(I) and the closer RA(I) is to 1, the better A performs on input I. Algorithm A has approximation ratio RA if RA RA(I) for all instances I of.

25 Next Class More on approximation algorithms How to show hardness of approximation Reading: Arora-Barak, 18.1, 18.2