CS 3133, A Term 1998
Foundations of Computer Science
Homework 5 (due Oct. 13)

This page is located at http://www.cs.wpi.edu/~alvarez/CS3133/HW5.html

Problems

1. Big O notation for two parameters: Given functions f(m,n) and g(m,n), the notation f(m,n) = O(g(m,n)) means that there are three numbers C, m₀, n₀ such that |f(m,n)| <= C |g(m,n)| whenever one simultaneously has both m>=m₀ and n>=n₀.
   Show that if a function f(m,n) satisfies f(m,n) <= n + m², then also f(m,n) = O(n m²). Your solution should include values for C, m₀, n₀ and an explanation of why those values work.

2. a) Construct a polynomial time verifier for the satisfiability problem 3-SAT discussed in class. Here, by a polynomial time verifier we mean a (deterministic) algorithm that decides, given both a formula H of the special form described below and a candidate truth assignment A, if A makes H true or not, and does so in time O(n^p m^q), where n is the total number of variables and m is the number of clauses in H.

   Recall that the input for the 3-SAT problem is a formula H in conjunctive normal form, i.e. a conjunction of m disjunctive clauses C₁, ...,C_m. Each clause C_i is the disjunction of exactly 3 literals L_i,1, L_i,2, L_i,3. Each literal is a Boolean variable x_j or its negation ~x_j. There are assumed to be n variables x₁,..., x_n. Different clauses may share some or all of their variables. The output for the 3-SAT problem is a yes/no answer, where yes means that there is a truth assignment for all of the variables x_j that makes H true according to the usual truth tables for negation (not), conjunction (and), and disjunction (or). A no answer means that every truth assignment for the variables makes H false.

   b) Carefully calculate the time complexity of your verifier as a function of the parameters m and n described in part a) above.

3. Extra credit:

   Show that 3-SAT can be reduced to the following vertex cover problem VC in polynomial time. This shows that VC is also NP-complete.

   The input to VC is a graph G and a number k. The output should be yes if there is a set C containing k or fewer vertices of G such that every edge of G has one of its endpoints in C. The output should be no otherwise.

   Hint: VC is related to the independent vertex set problem IVS discussed in class, in which a graph G and a number k are given, and it must be determined if there is a set I containing k vertices of G, no two of which are connected by an edge of G.