7.5 More NP-complete problems

















What known-NP-complete language is often used
to show another language is NP-complete?

	3SAT

What general approach is used to convert a 3SAT problem instance
into an instance of another problem?

	look for gadgets in the new problem
	gadgets are structures that simulate variables and clauses

What gadgets are used in the reduction from 3SAT to CLIQUE?

	individual nodes simulate variables/literals
	triples of nodes simulate clauses


















The Vertex Cover Problem


What's a Vertex Cover of an undirected graph G?

	a subset of the nodes of G, such that
	every edge of G touches one of the selected nodes

What's the Vertex Cover problem?

	VERTEX-COVER = { <G,k> | G is an undirected graph 
				with a k-node vertex cover }

















VERTEX-COVER is NP-Complete

How do you show VERTEX-COVER is NP-Complete?

	1. Show VERTEX-COVER is in NP
	2. Show 3SAT reduces to VERTEX-COVER

How do you show VERTEX-COVER is in NP?

	given a subset of nodes
	check that each edge has an endpoint in the set
	at most n^2 edges to check
	finding a node in the subset is no worse than O(n)
















3SAT Reduces to VERTEX-COVER


Given a 3SAT instance <F>, create a VERTEX-COVER instance <G,k>

What gadget is used for variables?

	a variable gadget is an edge and two nodes
	the nodes are labeled with 'x' and 'not x'
	setting x true corresponds to selecting x in the vertex cover

What gadget is used for clauses?

	clause gadgets are triples of three nodes
	the triples are connected with three edges
	the nodes are labeled with the literals from the clause

How are the variable and clause gadgets interconnected?

	connect the nodes in each triple to the variables with the same label

Construct the graph for the formula F.

	F = (not a or b or not c) and (a or b or d)


How do you show the construction is polynomial-time?
If the 3SAT formula has v variables and c clauses,
how many nodes and edges are created in the graph?

	2v + 3c nodes
	v + 6c edges

















What's the correct value of k for the VERTEX-COVER instance?

	set k to v + 2c


If F is satisfiable, how do you know G has a cover of size k?

	in each variable-gadget, add the node with a true label to the cover
	in each clause-gadget, select a node with a true label
	    add the two remaining nodes to the cover
	the cover has a total of k nodes

	the variable-gadget edges are covered
	the clause-gadget edges are covered
	the edges between gadgets are covered


If G has a cover of size k, how do you know F is satisfiable?

	the cover must contain one node from each variable gadget
		(its the only way to cover the edge in the variable gadget)
	the cover must contain two nodes from each clause gadget
		(its the only way to cover the three edges in clause gadget)
	this accounts for all the nodes in the vertex cover
	assign the literals true for the variable-gadget nodes in the cover

	F is satisfied
		only two interconnecting edges are covered by clause nodes
		the third must be covered by a variable node
		that variable makes the clause true