6520: CS Computational Complexity
Problem Set 1 Due March 4, 2008 Problem 1
Give a Karp reduction from CLIQUE to SAT.
Problem 2
Let Quadratic be the problem of deciding whether a given system of quadratic multivariate polynomial equations with integer coecients has a solution modulo 2. Prove that Quadratic is NP-Complete.
Problem 3
An NP minimization problem is dened by an objective function Obj : N for which there is a constant c and an algorithm that for every x, y {0, 1} , computes Obj(x, y) in time O(|x|c ). For such an objective function Obj, the corresponding NP minimization problem is: Given x , nd a y such that Obj(x, y) is minimized. Prove that P = NP if and only if every NP minimization problem has a polynomial-time algorithm.1
Problem 4
Prove that for each i N, i -SAT is P -Complete. i
Problem 5
Consider the Factoring problem: Given a natural number N , express N as a product of its prime factors. To date no polynomial-time algorithm for Factoring is known, and the conjectured hardness of Factoring has been the basis of several public-key cryptosystems. 1. Prove that if P = NP coNP, then Factoring can be solved by a polynomial-time algorithm. You may nd the following facts useful:
An NP maximization problem can be dened similarly, for which an analogous result holds.
1
1
The Fundamental Theorem of Arithmetics: Every integer greater than one can be decomposed into a product of prime factors, and moreover such a decomposition is unique up to the order of the prime factors. There is a polynomial-time algorithm for deciding whether a given integer is a prime. 2. Prove that unless NP = coNP, Factoring is not NP-Hard under Cook reduction. Conclude that unless NP = PH, Factoring is not NP-Hard under Cook reduction.
Problem 6
A language L is self-reducible if there is a oracle polynomial-time machine M such that (a) M L decides L, and (b) on every input x, M only makes queries of length strictly smaller than |x|. For instance, as shown in class, SAT and TQBF are self-reducible. Prove that every self-reducible language is in PSPACE.
Problem 7
1. A directed graph G = (V, E) is strongly connected if for every pair of vertices u, v V , there is a path from u to v in G. Prove that the problem of deciding whether a given directed graph is strongly connected is NL-Complete. 2. Prove that 2-SAT is NL-Complete. 3. Let Bipartite be the language consisting of bipartite graphs. Prove that Bipartite NL. 4. Let USTCONN be the following problem: Given an undirected graph G = (V, E) and two vertices s, t V , decide whether there is a path from s to t in G. Prove that USTCONN and Bipartite are computationally equivalent under log-space reductions.
2
Problem 8
1. Let s(n) be a space-constructible function. Prove that a language L NSPACE(s(n)) if and only if there is a deterministic Turing machine M with a read-once2 certicate tape such that for every x ...