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Chapter 9: Problem 10

Show that the reduction of the CNF-Satisfiability Problem to the Clique Decision Problem can be done in polynomial time.

Short Answer

Expert verified
Yes, the reduction of the CNF-Satisfiability Problem to the Clique Decision problem can be done in polynomial time. This is because the process of creating a graph from a CNF formula, identifying the cliques within this graph, and connecting vertices to not correspond to contradictory literals in other cliques can all be completed within polynomial time.

Step by step solution

01

Establish understanding of the CNF-Satisfiability and Clique Decision Problem

Firstly, ensure an understanding of both CNF-Satisfiability and Clique Decision Problems. CNF-Satisfiability refers to determining if there is an interpretation that satisfies a given Boolean formula in conjunctive normal form (CNF). The Clique Decision problem involves determining whether a graph contains a clique (a subset of vertices that forms a complete subgraph) of a particular size.
02

Create a graph from a CNF formula

Create a graph representation of the given CNF formula where each variable in the CNF formula corresponds to a vertex in the graph. For every clause in the CNF formula, create a clique in the graph of size equal to the number of literals in that clause. Connect every vertex in one of these cliques to all vertices that do not correspond to a contradictory literal in other cliques.
03

Identify the cliques

Identify the cliques in the created graph. These cliques represent the satisfiability of the original CNF formula.
04

Show the Reduction is Polynomial Time

To show that this entire process can be done in polynomial time, observe that the construction of the graph and the identification of the cliques can both be done in polynomial time with respect to the size of the original CNF formula.

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Most popular questions from this chapter

List three problems that have polynomial-time algorithms. Justify your answer.

Show that a problem is \(N P\) -easy if and only if it reduces to an \(N P\) -complete problem.

Can you develop an approximation algorithm for the CNF-Satisfiability Problem by stating it as an optimization problem- that is, by finding a truth assignment of the literals in the expression that makes the maximum possible number of clauses true?

Write a polynomial-time verification algorithm for the Hamiltonian Circuits Decision Problem.

Is the Towers of Hanoi Problem an NP-complete problem? Is it an \(N P\) -easy problem? Is it an \(N P\) -hard problem? Is it an \(N P\) -equivalent problem? Justify your answers. This problem is presented in Exercise 17 in Chapter 2.
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