CS510: ANALYSIS of ALGORITHMS
Kalamazoo College

Spring 1998

Satisfiability Introduction

Satisfiability -

Input: A set of Boolean variables V and a set of clauses C over V.

Output: Does there exist a satisfying truth assignment for C, i.e. a way to set the variables v₁, v₂, ..., v_n either true or false so that each clause contains at least one true literal?
- Example:
  1. A satisfiable example:
    The solution:
  2. a non-satisfiable example:
    The solution: None exists.
Theory of NP-Complete - Definitions:
- A problem is polynomial (or in the class P) if it can be solved in time polynomial in its size.
- A problem is nondeterministically polynomial (or in the class NP) if a conjectured answer can be verified in time polynomial in its size.
- Satisfiability is at least as hard as any problem in NP. Thus, if we can find a polynomial time algorithm for satisfiability, we can find a polynomial time algorithm for all problems in NP. (proof is outside the scope of this class)
- A problem is NP-hard if it is at least as hard as any problem in NP.
- A problem is NP-complete if it is NP-hard and also in NP itself.
- Thus, to show a problem is NP-complete you must:
  1. Verify that the problem is a decision problem and in NP.
  2. Find a translation from a known NP-complete problem instance to an instance of your problem.
  3. Show that every solution to the NP-complete problem is a solution to the corresponding instance of your problem and every solution to an instance of your problem is a solution to the corresponding NP-complete problem.

Example of showing a problem is NP-complete: 3-Satisfiability

Problem Description
- Input: A collection of clauses C where each clause contains exactly 3 literals over a set of Boolean Variables V.
- Output: Is there a truth assignment to V such that each clause is satisfied?
Example

Transformation of SAT to 3-SAT:

SAT CLAUSE	3-SAT CLAUSES
{ z₁ }	{ v₁, v₂, z₁ }, {v₁, v₂', z₁ }, {v₁', v₂, z₁ }, {v₁', v₂', z₁ }
{ z₁, z₂ }	{ v₁, z₁, z₂ }, {v₁', z₁, z₂ }
{ z₁, z₂, z₃ }	No Change Needed
{ z₁, z₂, ... , z_n }	{ v_j-1, z_j, v_j' } for 3 <= j <= n-2, {z₁, z₂, v₂' }, {v_n-2, z_n-1, z_n }

For each transformation given above, we need to check that one or more z_i's being true gives that at least one SAT clause is true and at least one SAT clause being true gives one or more z_i's must be true.
Transformation takes O(m+n) time where m = number of literals and n = number of clauses.

Summary of Satisfiability:
- 1-SAT is always recognizable. What's an algorithm?
- 2-SAT is solvable in polynomial time. See Section 8.3.10.
- k-SAT is NP-Complete for k >= 3. The reduction is similar to the one for 3-SAT.

Kelly Schultz, Kalamazoo College