Science Fair Project Encyclopedia
Independent set problem
Given a graph G, an independent set is a subset of its vertices that are pairwise not connected together.
- A graph G
- An integer K.
- Does G have an independent set of at least size K?
Proof that Independent Set is NP-complete
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)
- 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
- 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.
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