CS254 Lecture 7 – Valiant-Vazirani

Today we prove the Valiant-Vazirani theorem.

Theorem 1 (Valiant-Vazirani) Suppose there is a polynomial time algorithm that on input a CNF formula having exactly one satisfying assignment finds that assignment. (We make no assumption on the behaviour of the algorithm on other inputs.) Then {{\bf NP}={\bf RP}}.

1. The Valiant-Vazirani Theorem

As discussed in the last lecture, our approach is the following: given a satisfiable formula {\phi} and a number {k} such that {\phi} has roughly {2^k} satisfying assignments, we pick a random hash function {h: \{ 0,1 \}^n \rightarrow \{ 0,1 \}^{k+2}} from a family of pairwise independent hash functions, and we construct a formula {\psi(x)} which is equivalent to {\phi(x) \wedge (h(x) = 0)}. With constant probability, {\psi} has precisely one satisfying assignment, and so we can pass it to our hypothetical algorithm, which finds a satisfying assignment for {\psi} and hence a satisfying assignment for {\phi}.

If we are only given {\phi}, we can try all possible values of {k} between {0} and {n} (where {n} is the number of variables in {\phi}), and run the above procedure for each {k}. When the correct value of {k} is chosen, we have a constant probability of finding a satisfying assignment for {\phi}.

Once we have a randomized algorithm that, given a satisfiable formula, finds a satisfying assignment with constant probability, we have an {{\bf RP}} algorithm for 3SAT: run the assignment-finding algorithm, accept if it finds a satisfying assignment and reject otherwise. The existence of an {{\bf RP}} algorithm for 3SAT implies that {{\bf NP} \subseteq {\bf RP}} because {{\bf RP}} is closed under many-to-one reductions, and so {{\bf RP} = {\bf NP}} because we have {{\bf RP} \subseteq {\bf NP}} by definition.

The main calculation that we need to perform is to show that if we have a set of size roughly {2^k}, and we hash its elements pairwise independently to {\{ 0,1 \}^{k+2}}, then there is a constant probability that exactly one element is hashed to {(0,\ldots,0)}.

Lemma 2 Let {T\subseteq \{ 0,1 \}^n} be a set such that {2^k \leq |T| < 2^{k+1}} and let {H} be a family of pairwise independent hash functions of the form {h:\{ 0,1 \}^n \rightarrow \{ 0,1 \}^{k+2}}. Then if we pick {h} at random from {H}, there is a constant probability that there is a unique element {x\in T} such that {h(x) = {\bf 0}}. Precisely,

\displaystyle  \mathop{\mathbb P}_{h\in H} [ | \{ x \in T : h(x)={\bf 0} \}|=1] \geq \frac 18

Proof: Let us fix an element {x\in T}. We want to compute the probability that {x} is the unique element of {T} mapped into {\bf 0} by {h}. Clearly,

\displaystyle  \mathop{\mathbb P}_h[ h(x)= {\bf 0} \wedge \forall y\in T-\{x\}. h(y) \neq {\bf 0}] = \mathop{\mathbb P}_h[h(x)= {\bf 0}] \cdot \mathop{\mathbb P}_h [ \forall y\in T-\{x\}. h(y) \neq {\bf 0}| h(x)= {\bf 0}]

and we know that

\displaystyle  \mathop{\mathbb P}_h[h(x)= {\bf 0}] = \frac 1 {2^{k+2}}

The difficult part is to estimate the other probability. First, we write

\displaystyle  \mathop{\mathbb P}_h [ \forall y\in T-\{x\}. h(y) \neq {\bf 0}| h(x)= {\bf 0}] = 1 - \mathop{\mathbb P}_h [ \exists y\in T-\{x\}. h(y) = {\bf 0}| h(x)= {\bf 0}]

And then observe that

\displaystyle  \begin{array}{rcl}  & & \mathop{\mathbb P}_h [ \exists y\in T-\{x\}. h(y) = {\bf 0}| h(x)= {\bf 0}] \\ & \leq & \sum_{y\in |T|-\{x\}} \mathop{\mathbb P}_h [ h(y) = {\bf 0}| h(x)= {\bf 0}] \\ &=& \sum_{y\in |T|-\{x\}} \mathop{\mathbb P}_h [ h(y) = {\bf 0}]\\ & = & \frac {|T|-1}{2^{k+2}}\\ & \leq & \frac 12 \end{array}

Notice how we used the fact that the value of {h(y)} is independent of the value of {h(x)} when {x\neq y}.

Putting everything together, we have

\displaystyle  \mathop{\mathbb P}_h [ \forall y\in T-\{x\}. h(y) \neq {\bf 0}| h(x)= {\bf 0}] \geq \frac 12

and so

\displaystyle  \mathop{\mathbb P}_h[ h(x)= {\bf 0} \wedge \forall y\in T-\{x\}. h(y) \neq {\bf 0}] \geq \frac 1 {2^{k+3}}

To conclude the argument, we observe that the probability that there is a unique element of {T} mapped into {\bf 0} is given by the sum over {x\in T} of the probability that {x} is the unique element mapped into {\bf 0} (all this events are disjoint, so the probability of their union is the sum of the probabilities). The probability of a unique element mapped into {\bf 0} is then at least {|T|/2^{k+3} > 1/8}. \Box

Lemma 3 There is a probabilistic polynomial time algorithm that on input a CNF formula {\phi} and an integer {k} outputs a formula {\psi} such that

  • If {\phi} is unsatisfiable then {\psi} is unsatisfiable.
  • If {\phi} has at least {2^k} and less than {2^{k+1}} satifying assignments, then there is a probability at least {1/8} then the formula {\psi} has exactly one satisfying assignment.

Proof: Say that {\phi} is a formula over {n} variables. The algorithm picks at random vectors {a_1,\ldots,a_{k+2}\in \{ 0,1 \}^n} and bits {b_1,\ldots,b_{k+2}} and produces a formula {\psi} that is equivalent to the expression {\phi(x) \wedge (a_1\cdot x +b_1=0) \wedge \ldots \wedge (a_{k+2} \cdot x +b_{k+2}=0)}. Indeed, there is no compact CNF expression to compute {a\cdot x} if {a} has a lot of ones, but we can proceed as follows: for each {i} we add auxiliary variables {y^i_1,\ldots,y^i_n} and then write a CNF condition equivalent to {(y^i_1 = x_1 \wedge a_{i}[1] ) \wedge \cdots \wedge (y^i_n = y^i_{n-1} \oplus (x_{n} \wedge a_{i}[n] \oplus b_i)))}. Then {\psi} is the AND of the clauses in {\phi} plus all the above expressions for {i=1,2,\ldots,k+2}.

By construction, the number of satisfying assignments of {\psi} is equal to the number of satisfying assignments {x} of {\phi} such that {h_{a_1,\ldots,a_{k+2},b_1,\ldots,b_{k+2}} (x)={\bf 0}}. If {\phi} is unsatisfiable, then, for every possible choice of the {a_i}, {\psi} is also unsatisfiable.

If {\phi} has between {2^k} and {2^{k+1}} assignments, then Lemma 2 implies that with probability at least {1/8} there is exactly one satisfying assignment for {\psi}. \Box

We can now prove the Valiant-Vazirani theorem.

Proof: (Of Theorem 1) It is enough to show that, under the assumption of the Theorem, 3SAT has an {{\bf RP}} algorithm.

On input a formula {\phi}, we construct formulae {\psi_0,\ldots,\psi_n} by using the algorithm of Lemma 3 with parameters {k=0,\ldots,n}. We submit all formulae {\psi_0,\ldots,\psi_{n}} to the algorithm in the assumption of the Theorem, and accept if the algorithm can find a satisfying assignment for at least one of the formulae. If {\phi} is unsatisfiable, then all the formulae are always unsatisfiable, and so the algorithm has a probability zero of accepting. If {\phi} is satisfiable, then for some {k} it has between {2^k} and {2^{k+1}} satisfying assignments, and there is a probability at least {1/8} that {\psi_{k}} has exactly one satisfying assignment and that the algorithm accepts. If we repeat the above procedure {t} times, and accept if at least one iteration accepts, then if {\phi} is unsatisfiable we still have probability zero of accepting, otherwise we have probability at least {1- (7/8)^t} of accepting, which is more than 1/2 already for {t=6}. \Box

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One thought on “CS254 Lecture 7 – Valiant-Vazirani

  1. Thanks for the post.

    Could you please explain the part where you have used auxiliary variables.
    How is the CNF expression with these auxiliary variables equivalent to the earlier CNF expression, that is, (a_1.x+b_1=0) … ?

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