Complexity

1
Complexity
11-1
NP-Completeness
Complexity Andrei Bulatov
2
Complexity
11-2
NP-Completeness Proofs
To prove that a language L is NP-complete we
now just have to perform two steps

• Show that L belongs to NP
• Find a known NP-complete problem (language) L?
  and show L? ? L

100s of problems have now been shown to be
NP-complete (for an earlier survey see Garey and
Johnson)
Note If we can complete Step 2 but not Step
1, then we say that L is NP-hard
3
Complexity
11-3
NP-Completeness of Clique
Theorem Clique is NP-complete
Step 1 The problem Clique is in NP
the list of vertices
in the clique is the certificate
4
Complexity
11-4
Step 2 To show that Clique is
NP-complete we shall reduce Satisfiability
to Clique
This construction can be carried out in
polynomial time
The resulting graph has a clique of size k if
and only if ? is satisfiable (assign the value
true to every variable occurring in the clique)
5
Complexity
11-5
Example Construction
k ? 3
6
Complexity
11-6
NP-Completeness of Vertex Cover
Step 1 The problem Vertex Cover is in
NP the list of
vertices in M is the certificate
7
Complexity
11-7
Step 2 To show that Vertex Cover is
NP-complete we shall reduce
Satisfiability to Vertex Cover
This construction can be carried out in
polynomial time
8
Complexity
11-8
Example Construction
X
9
Complexity
11-9
10
Complexity
11-10
Alternative Reductions
We have shown that Clique and Vertex Cover
are both NP-complete
This means it must be possible to reduce Clique
to Vertex Cover and vice versa!
11
Complexity
11-11
NP-Completeness of 3-SAT
To show that 3-Satisfiability is NP-complete we
reduce Satisfiability to 3-Satisfiability

• C is satisfiable if and only if C? is,
  since at least one of the literals
• other than Ys must be true

12
Complexity
11-12
NP-Completeness of SubsetSum
Step 1 The problem SubsetSum is in
NP the set T is
the certificate
13
Complexity
11-13
Step 2 To show that SubsetSum is
NP-complete we shall reduce
Satisfiability to SubsetSum

• Choose t so that T must contain exactly
  one of each pair
• and at least one from each clause

This construction can be carried out in
polynomial time
14
Complexity
11-14
Example Construction
15
Complexity
11-15
NP-Completeness of HamCircuit
Theorem HamCircuit is
NP-complete
Step 1 The problem HamCircuit is in
NP the Hamilton
circuit is the certificate
16
Complexity
11-16
Step 2 To show that HamCircuit is
NP-complete we shall reduce
3-Satisfiability to HamCircuit
What is to be encoded?

• Boolean variables
• a choice between two values (for each
  variable)
• consistency all occurrences of X must
  have the same truth value
• constraints on the possible values imposed by
  clauses

17
Complexity
11-17

• the choice gadget

• We assume that all gadgets are connected
• with the rest of the graph only through
  their
• endpoints, shown as full dots there are no
• edges connecting other vertices of the
  gadget
• to the rest of the graph

• This gadget will allow the Hamiltonian
  circuit,
• approaching from above, to pick either left
• or right edge, thus communicating to a
  truth
• value

18
Complexity
11-18

• the consistency gadget

• This graph can be traversed by the
  Hamiltonian circuit in one of the
• two ways

19
Complexity
11-19

• the constraint gadget

• If, using the choice and consistency devices,
  we have made sure that each
• side of the triangle is traversed by the
  Hamilton circuit if and only if the
• corresponding literal is false then at
  least one literal has to be true

20
Complexity
11-20
Properties of the Gadgets

• the choice gadget can be traversed in exactly
  two ways
• the internal vertices of the consistency
  gadget (exclusive or gadget)
• can be traversed in exactly two ways, so
  that exactly one pair of the
• external vertices is involved
• any Hamilton circuit traverses at most two of
  the edges of a
• constraint gadget

21
Complexity
11-21
X
Y
Z
2
1
22
Complexity
11-22
NP-Completeness of TSP(D)
Step 1 The problem TSP(D) is in NP
a route satisfying the
inequality is the certificate
23
Complexity
11-23
Step 2 To show that TSP(D) is
NP-complete we shall reduce HamCircuit
to TSP(D)
Given a graph G with vertex set V and edge
set E

• For each vertex v create a city
• Set if (u,v) ? E
  and otherwise
• Set B V

Then

• If G has a Hamilton circuit then there is
  a route of weight B
• (the Hamilton circuit)
• If there is a route of weight B, then in G
  this route goes through
• edges and therefore is a Hamilton circuit