The Vertex Cover Problem Theorem 7.44 Pavan Kantharaju Drexel - PowerPoint PPT Presentation

The Vertex Cover Problem Theorem 7.44 Pavan Kantharaju Drexel University - Department of Computer Science March 13, 2015

Reduction Definition (7.28) A function f : Σ ∗ → Σ ∗ is a polynomial time computable function if some polynomial time Turing Machine M exists that halts with just f ( w ) on its tape,when started on any input w . Definition (7.29) Language A is polynomial time reducible to language B , written as A ≤ P B , if a polynomial time computable function exists, where for every w , w ∈ A ⇐ ⇒ f ( w ) ∈ B

NP-Complete Definition (7.34) A language B is NP-Complete if it satisfies the following conditions: 1. B is in NP 2. Every A in NP can be polynomial-time reducible to B Theorem (7.36) Given that A is NP-complete and B in NP, if A ≤ P B, then B is NP-complete.

Vertex Cover Problem Definition (vertex cover) Given a graph G of vertices V and edges E , find a subset of nodes G ′ ⊆ G such that nodes in G ′ touch (or cover) all edges E in G . VERTEX-COVER = {� G , k �| G is an undirected graph that has a k -node vertex cover } To show that VERTEX-COVER is NP-complete, we show that the problem satisfies the constraints in Definition 7.34.

Vertex-cover is in NP Proof We can use a vertex cover of size k as the certificate to verify our solution. The following is a verifier V for VERTEX-COVER V =”On input �� G , k � , c � : 1. Test whether c is a subgraph with k nodes in G 2. Test whether c covers all edges in G 3. If yes, accept. If no, reject.”

3 − SAT ≤ P VERTEX − COVER Proof - Generating a graph First, we define a gadget . A gadget is a structure that simulates variables and clauses of a satisfiability formula. 3-CNF-SAT = {� φ �| φ is a 3-CNF satisfiable Boolean formula } Definition (Variable gadget) For each variable x in φ , produce two nodes x and ¯ x , and connect them with an edge. Definition (Clause gadget) For each clause in φ , create three nodes, labeled by the literals of the clause. All nodes are connected to each other. Finally, we connect each node in the clause gadget with its identical labels in the variable gadget. This will complete the graph.

3 − SAT ≤ P VERTEX − COVER Proof - Value of k Let m and l be the number of variables and clauses , respectively. Thus, we have 2 m + 3 l nodes in G . We let k = m + 2 l .

Example Reduction Sipser, Page 313 φ = ( x 1 ∨ x 1 ∨ x 2 ) ∧ ( ¯ x 1 ∨ ¯ x 2 ∨ ¯ x 2 ) ∧ ( ¯ x 1 ∨ x 2 ∨ x 2 ) Reduction produces � G , k � from φ , where k = 8. G is the following: Figure : 7.45

3 − SAT ≤ P VERTEX − COVER Proof - Correctness of reduction φ is satisfiable if and only if G has a vertex cover of size k . If φ is satisfiable, then G has a vertex cover of size k . 1. For each variable gadget, place the variable corresponding to the true literal in the assignment into the vertex cover. 2. For each clause gadget, select one true literal and put the remaining two nodes into the vertex cover. 3. Vertex cover contains k nodes, covering all edges in G .

3 − SAT ≤ P VERTEX − COVER Proof - Correctness of reduction φ is satisfiable if and only if G has a vertex cover of size k . If G has a vertex cover of size k , then φ is satisfiable. The vertex cover must contain: ◮ One node from each variable gadget ◮ Two nodes from each clause gadget Take the nodes of the variable gadget that are in the vertex cover and assign true to each literal.

Thus, VERTEX-COVER is NP-Complete.

Questions?