Question

When all instances of the CNF-Satisfiability Problem have exactly three literals per clause, it is called the 3 -Satisfiability Problem. Knowing that the 3-Satisfiability Problem is \(N P\) -complete, show that the Graph 3 -Coloring Problem is also \(N P\) -complete.

Short answer

The Graph 3-Coloring Problem is NP-Complete as we have shown a reduction from the 3-Satisfiability Problem, which is already known to be NP-complete, to the Graph 3-Coloring Problem. We have also verified that the Graph 3-Coloring Problem is in NP, hence, it is NP-Complete.

Step-by-step solution

Definition of CNF 3-Satisfiability Problem

A boolean formula is in Conjunctive Normal Form (CNF) if it is a conjunction of clauses, where each clause is a disjunction of literals. The 3-Satisfiability Problem (3-SAT) is a decision problem where you are given a CNF formula with exactly 3 literals per clause, and the question is to determine whether there exists some assignment of true and false values to the variables that would make the entire formula true.

Definition of Graph 3-Coloring Problem

The Graph 3-Coloring Problem is a decision problem where you are given a graph, and the question is to determine whether it is possible to assign one of 3 colors to each vertex, such that no two adjacent vertices share the same color.

Describe a reduction from 3-SAT to Graph 3-Coloring

We know that 3-SAT is NP-complete. To show that Graph 3-Coloring is also NP-Complete, we need to perform a reduction from 3-SAT to Graph 3-Coloring. Reduction from 3-SAT to Graph 3-Coloring involves creating a graph based on the CNF formula such that solving Graph 3-Coloring problem on that graph is equivalent to solving the 3-SAT problem on the CNF formula. Specifically, construct a graph where each clause in the CNF formula corresponds to a unique subgraph in the graph. Each variable in the CNF formula corresponds to a pair of vertices in the graph connected by an edge. Moreover, those vertices are connected to the vertices in the subgraphs that correspond to the occurrences of the variable or its negation in the CNF formula. If it's possible to properly 3-color the constructed graph, there exists a satisfying assignment for the CNF formula.

Proving NP-Completeness of Graph 3-Coloring

By outline in Step 3, we have shown a reduction from 3-SAT to Graph 3-Coloring. It's clear that the Graph 3-Coloring problem is in NP, because given a 3-coloring of a graph, we can easily check in polynomial time that no two adjacent vertices share the same color. This reduction proves that if we could solve the Graph 3-Coloring problem in polynomial time, we could also solve the 3-SAT problem, which is known to be NP-Complete, in polynomial time. Therefore, the Graph 3-Coloring is NP-Complete

Key concepts

3-Satisfiability Problem

The 3-Satisfiability Problem, often abbreviated as 3-SAT, is a classic problem in computer science. It is part of a larger class of problems known as NP-Complete. In simple terms, the 3-SAT problem involves a Boolean formula in Conjunctive Normal Form (CNF) and requires determining if there is a way to assign truth values—true or false—to its variables such that the entire formula evaluates to true.

A CNF formula consists of clauses joined together with an AND operation. Each clause within this CNF formula is made up of exactly three literals, which are either a variable or its negation, combined using an OR operation. The challenge is to find an assignment of variables that makes every clause true simultaneously. Solving the 3-SAT problem has significant implications in various fields, including logic, cryptography, and algorithm design.

Graph 3-Coloring Problem

The Graph 3-Coloring Problem is another well-known NP-Complete problem. The objective is to determine if it's possible to color the vertices of a given graph using no more than three colors. Importantly, adjacent vertices, or vertices connected by an edge, must not be colored the same.

• A perfect 3-coloring ensures different colors for adjacent vertices.
• This constraint makes the problem both interesting and challenging.

This problem holds practical significance in network design, scheduling, and resource allocation. Showing that the Graph 3-Coloring Problem is NP-Complete often involves reducing another already established NP-Complete problem to it, such as the 3-SAT problem.

Boolean formula

A Boolean formula is a mathematical expression involving Boolean variables and operators. The main operators in Boolean algebra are AND, OR, and NOT.

• Boolean variables can only take values true or false (1 or 0).
• Boolean formulas are essential in fields like computer science, electronics, and mathematical logic.

One important use of Boolean formulas is in the creation of logical statements seen in problems like 3-SAT. These logical expressions are fundamental in constructing algorithms, designing circuits, and performing data searches.

Conjunctive Normal Form (CNF)

Conjunctive Normal Form, commonly known as CNF, is a way of organizing a Boolean formula as a conjunction of disjunctions. In simpler terms, it's an expression consisting of ANDs of ORs. Each clause in a CNF formula is made up of literals, connected by OR operators, whereas the overall formula combines these clauses using AND operators.

• A literal is a variable or the negation of a variable.
• CNF is fundamental in many computational problems, including the 3-SAT problem.

The structure of CNF makes it particularly useful for proving concepts related to NP-Completeness and for developing efficient algorithms in logic and computation.