CS 6520: 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 coefficients has
a solution modulo 2. Prove that Quadratic is NP-Complete.
Problem 3
An NP minimization problem is defined by an objective function Obj :
Σ∗×Σ∗→Nfor which there is a constant cand 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∈Σ∗,
find a y∈Σ∗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, Σi-SAT is ΣP
i-Complete.
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 find the following facts useful:
1An NP maximization problem can be defined similarly, for which an analogous result
holds.
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