CS614. Advanced Algorithms. L20 Quiz.

CS614. Advanced Algorithms.

L20 Quiz

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Problem 1. Monotone Satisfiability with Few True Variables

Consider an instance of the Satisfiability Problem, specified by clauses $C_1, \ldots, C_k$ over a set of Boolean variables $x_1, \ldots, x_n$. We say that the instance is monotone if each term in each clause consists of a nonnegated variable; that is, each term is equal to $x_i$, for some $i$, rather than $\bar{x}_i$. Monotone instances of Satisfiability are very easy to solve: They are always satisfiable, by setting each variable equal to 1 . For example, suppose we have the three clauses $$\left(x_1 \vee x_2\right),\left(x_1 \vee x_3\right),\left(x_2 \vee x_3\right) .$$ This is monotone, and indeed the assignment that sets all three variables to 1 satisfies all the clauses. But we can observe that this is not the only satisfying assignment; we could also have set $x_1$ and $x_2$ to 1 , and $x_3$ to 0 . Indeed, for any monotone instance, it is natural to ask how few variables we need to set to 1 in order to satisfy it.

Given a monotone instance of Satisfiability, together with a number $k$, the problem of Monotone Satisfiability with Few True Variables asks: Is there a satisfying assignment for the instance in which at most $k$ variables are set to 1?

Prove this problem is NP-complete. Hint: reduce from vertex cover.

Problem 2. ALL-or-NOTHING-3SAT

The problem ALL-or-NOTHING-3SAT asks, given a 3CNF boolean formula, whether there is an assignment to the variables such that each clause either has three TRUE literals or has three FALSE literals.

1. Describe a polynomial-time algorithm to solve ALL-or-NOTHING-3SAT.

2. But 3SAT is NP-hard! Why doesn’t the existence of this algorithm prove that P=NP?