Independent set problem

In mathematics, the independent set problem (IS) is a question in combinatorics, known to be an NP-complete problem.

Description

Given a graph G, an independent set is a subset of its vertices that are pairwise not connected together.

Input:

• A graph G
• An integer K.

Question:

• Does G have an independent set of at least size K?

Proof that Independent Set is NP-complete

This is by reduction from 3-SAT, a version of the Boolean satisfiability problem.

1. Certificate: Check that no vertices are connected.

This can easily be verified in polynomial time.

2. 3-CNF-SAT->IS transformation

• 3-CNF-SAT (Given):
  • Variables x1, x2, ..., xn
  • Clauses c1, c2, ..., cm
• IS (Construction of Graph):
  • 1 vertex for each occurrence of each literal (3m vertices)
  • Connect two vertices when:
    • They are conflicting (i.e. x1, ~x1)
    • They are in the same clause

This transformation can be performed in polynomial time.

3. Proof of correctness (note: this is not formal or complete)

• Yes->Yes:
  • Pick only one literal from 3-CNF-SAT solution
  • Per definition of satisfiability, vertices in our graph represented by this literal will not be connected to another vertex in the ""same clause"" or to a conflicting vertex
  • Hence, the node is part of the Independent Set
• No->No:
  • Use contrapositive (Yes<-Yes aka 3-CNF-SAT<-IS).
  • Select an independent set of size m.
  • By construction, vertices are linked to conflicting literals and literals in the same clause.
  • Therefore, these can not be part of an independent set.
  • Hence, only vertices in an independent set will satisfy 3-CNF-SAT.