1. NP-Completeness: Reductions
CSc 4520/6520
Fall 2013
Slides adapted from David Evans (Virgina) and Costas Busch (RPI)

2. Definition of NP-Complete
Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time is reducible to Q

3. Getting Started
How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP-Complete problem and just reduce from that P1 P2
Mystery NP-Complete Problem
Reference halting problem and undecidability Q P3 P4

4. 3-SAT
3-SAT = { f | f is in Conjunctive Normal Form, each clause has exactly 3 literals and f is satisfiable } 3-SAT is NP-Complete (2-SAT is in P)

5. NP-Complete
To prove a problem is NP-Complete show a polynomial time reduction from 3-SAT
Other NP-Complete Problems:
PARTITION
SUBSET-SUM
CLIQUE
HAMILTONIAN PATH (TSP)
GRAPH COLORING
MINESWEEPER (and many more)

6. NP-Completeness Proof Method
To show that Q is NP-Complete: 1) Show that Q is in NP 2) Pick an instance, R, of your favorite NP-Complete problem (ex: Φ in 3-SAT) 3) Show a polynomial algorithm to transform R into an instance of Q

7. Example: Clique
CLIQUE = { <G,k> | G is a graph with a clique of size k } A clique is a subset of vertices that are all connected Why is CLIQUE in NP?

8. Reduce 3-SAT to Clique
Pick an instance of 3-SAT, Φ, with k clauses Make a vertex for each literal Connect each vertex to the literals in other clauses that are not the negation Any k-clique in this graph corresponds to a satisfying assignment

11. Example: Independent Set
INDEPENDENT SET = { <G,k> | where G has an independent set of size k } An independent set is a set of vertices that have no edges How can we reduce this to clique?

12. Independent Set to CLIQUE
This is the dual problem!

13. Vertex Cover
Vertex cover of a graph
is a subset of nodes such that every edge
in the graph touches one node in
Example:
S = red nodes

14. Size of vertex-cover
is the number of nodes in the cover
Example:
|S|=4

15. Corresponding language:
VERTEX-COVER = { :
graph contains a vertex cover
of size }
Example:

16. VERTEX-COVER is NP-complete
Theorem:
VERTEX-COVER is NP-complete
Proof:
1. VERTEX-COVER is in NP
Can be easily proven
2. We will reduce in polynomial time
3CNF-SAT to VERTEX-COVER
(NP-complete)

17. Let be a 3CNF formula with variables and clauses Example: Clause 1

18. Formula can be converted
to a graph such that:
is satisfied
if and only if
Contains a vertex cover
of size

19. Variable Gadgets nodes Clause Gadgets nodes Clause 1 Clause 2 Clause 3

20. Clause 1
Clause 2
Clause 3
Clause 2
Clause 3
Clause 1

21. First direction in proof:
If is satisfied,
then contains a vertex cover of size

22. Example:
Satisfying assignment
We will show that contains
a vertex cover of size

23. Put every satisfying literal in the cover

24. Select one satisfying literal in each clause gadget
and include the remaining literals in the cover

25. This is a vertex cover since every edge is
adjacent to a chosen node

26. Explanation for general case:
Edges in variable gadgets
are incident to at least one node in cover

27. Edges in clause gadgets
are incident to at least one node in cover,
since two nodes are chosen in a clause gadget

28. Every edge connecting variable gadgets
and clause gadgets is one of three types:
Type 1
Type 2
Type 3
All adjacent to nodes in cover

29. Second direction of proof:
If graph contains a vertex-cover
of size
then formula is satisfiable

30. Example:

31. To include “internal’’ edges to gadgets, and satisfy
exactly one literal in each variable gadget is chosen
chosen out of
exactly two nodes in each clause gadget is chosen
chosen out of

32. For the variable assignment choose the
literals in the cover from variable gadgets

33. is satisfied with
since the respective literals satisfy the clauses

34. Dominating Set Goal: Problem:
Dominating-set = {<G, K> | A dominating set of size K for G exists}
Goal:
Show that Dominating-set is NP-Complete

35. Dominating Set (Definition)
Problem:
Dominating-set = {<G, K> | A dominating set of size (at most) K for G exists}
Let G=(V,E) be an undirected graph
A dominating set D is a set of vertices that covers all vertices
i.e., every vertex of G is either in D or is adjacent to at least one vertex from D

36. Dominating Set (Example)
Size-2 example : {Yellow vertices} e

37. Dominating Set (Proof Sketch)
Steps:
Show that Dominating-set ∈ NP.
Show that Dominating-set is not easier than a NPC problem
We choose this NPC problem to be Vertex cover
Reduction from Vertex-cover to Dominating-set
Show the correspondence of “yes” instances between the reduction

38. Dominating Set - (1) NP
It is trivial to see that Dominating-set ∈ NP Given a vertex set D of size K, we check whether (V-D) are adjacent to D i.e., for each vertex, v, in (V-D), whether v is adjacent to some vertex u in D

39. Dominating Set - (2) Reduction
Reduction - Graph transformation
Construct a new graph G' by adding new vertices and edges to the graph G as follows: T G G’
<G,k> ∈ L
Vertex-cover
Dominating-set
<G’, k> ∈ L’

40. Dominating Set - (2) Reduction
Reduction - Graph transformation (Con’t)
For each edge (v, w) of G, add a vertex vw and the edges (v, vw) and (w, vw) to G'
Furthermore, remove all vertices with no incident edges; such vertices would always have to go in a dominating set but are not needed in a vertex cover of G T G G’
<G,k> ∈ L
Vertex-cover
Dominating-set
<G’, k> ∈ L’

41. Vertex cover
A vertex cover, C, is a set of vertices that covers all edges
i.e., each edge is at least adjacent to some node in C 1 2 3 4
{2, 4}, {3, 4}, {1, 2, 3}
are vertex covers

42. Dominating Set – Graph Transformation Examplew v z u vz wu vw zu vu G' v w z u G

43. Dominating Set - (3) Correspondence
A dominating set of size K in G’  A vertex cover of size K in G
 Let D be a dominating set of size K in G’
Case 1): D contains only vertices from G
Then, all new vertices have an edge to a vertex in D
D covers all edges
D is a valid vertex cover of G

44. Dominating Set - (3) Correspondence
A dominating set of size K in G’  A vertex cover of size K in G
 Let D be a dominating set of size K in G’
Case 2): D contains some new vertices (vertex in the form of uv)
(We show how to construct a vertex cover using only old vertices, otherwise we cannot obtain a vertex cover for G)
For each new vertex uv, replace it by u (or v)
If u ∈ D, this node is not needed
Then the edge u-v in G will be covered
After new edges are removed, it is a valid vertex cover of G (of size at most K)

45. Dominating Set - (3) Correspondence
A dominating set of size K in G’  A vertex cover of size K in G
 Let C be a vertex cover of size K in G
For an old vertex, v ∈ G’ :
By the definition of VC, all edges incident to v are covered
v is also covered
For a new vertex, uv ∈ G’ :
Edge u-v must be covered, either u or v ∈ C
This node will cover uv in G’
Thus, C is a valid dominating for G’ (of size at most K)

46. Dominating Set - (3) Correspondencevw v w v w vz wu vu z u z u zu
Vertex-cover in G
Dominating-set in G'