1. CS21 Decidability and Tractability
Lecture 23
March 6, 2019
March 6, 2019
CS21 Lecture 23

2. Outline NP-complete problem: max cut (finishing up)
The complexity class coNP
The complexity class coNP ∩ NP
The complexity class PSPACE
a PSPACE-complete problem
March 6, 2019
CS21 Lecture 23

3. MAX CUT Theorem: the following language is NP-complete: Proof:
MAX CUT = {(G = (V, E), k) : there is a cut S ⊆ V with at least k edges crossing it}
Proof:
Part 1: MAX CUT ∈ NP. Proof?
Part 2: MAX CUT is NP-hard.
reduce from?
March 6, 2019
CS21 Lecture 23

4. MAX CUT is NP-complete We are reducing from the language:
NAE3SAT = {φ : φ is a 3-CNF formula for which there exists a truth assignment in which every clause has at least 1 true literal and at least 1 false literal}
to the language:
MAX CUT = {(G = (V, E), k) : there is a cut S ⊆ V with at least k edges crossing it}
March 6, 2019
CS21 Lecture 23

5. (x1 ∨ x2 ∨¬x3)∧(¬x1 ∨ x4 ∨ x5)∧…∧(¬x2 ∨ x3 ∨ x3 )
MAX CUT is NP-complete
The reduction:
given instance of NAE3SAT (n nodes, m clauses):
(x1 ∨ x2 ∨¬x3)∧(¬x1 ∨ x4 ∨ x5)∧…∧(¬x2 ∨ x3 ∨ x3 )
produce graph G = (V, E) with node for each literal
¬x3 x4
¬x4
triangle for each 3-clause
parallel edges for each 2-clause x2
¬x1
¬x5 x1
¬x2 x5 x3
March 6, 2019
CS21 Lecture 23

6. (repeat variable in 2-clause to make 3-clause for this calculation)
MAX CUT is NP-complete
¬x3 x4
¬x4
triangle for each 3-clause
parallel edges for each 2-clause x2
¬x1
¬x5 x1
¬x2 x5 x3
if cut selects TRUE literals, each clause contributes 2 if NAE, and < 2 otherwise
need to penalize cuts that correspond to inconsistent truth assignments
add ni parallel edges from xi to ¬xi (ni = # occurrences)
(repeat variable in 2-clause to make 3-clause for this calculation)
March 6, 2019
CS21 Lecture 23

7. MAX CUT is NP-complete YES maps to YES triangle for each 3-clause
parallel edges for each 2-clause
ni parallel edges from xi to ¬xi
set k = 5m
¬x3 x4
¬x4 x2
¬x1
¬x5 x1
¬x2 x5 x3
YES maps to YES
take cut to be TRUE literals in a NAE truth assignment
contribution from clause gadgets: 2m
contribution from (xi, ¬xi) parallel edges: 3m
March 6, 2019
CS21 Lecture 23

8. MAX CUT is NP-complete NO maps to NO triangle for each 3-clause
parallel edges for each 2-clause
ni parallel edges from xi to ¬xi
set k = 5m
¬x4 x2
¬x1
¬x5 x1
¬x2 x3 x5
NO maps to NO
Claim: if cut has xi, ¬xi on same side, then can move one to opposite side without decreasing # edges crossing cut
contribution from (xi, ¬xi) parallel edges: 3m
contribution from clause gadgets must be 2m
conclude: there is a NAE assignment
March 6, 2019
CS21 Lecture 23

9. MAX CUT is NP-complete
Claim: if cut has xi, ¬xi on same side, then can move one to opposite side without decreasing # edges crossing cut
Proof: xi
a + ni
a edges
... xi
¬xi
ni edges
a+b ≤ 2ni xi
...
b + ni
¬xi
...
b edges
a + ni ≥ a + b or
b + ni ≥ a + b
¬xi
March 6, 2019
CS21 Lecture 23

10. coNP Is NP closed under complement? Can we transform this machine:
x ∉ L
Can we transform this machine:
x ∈ L
x ∈ L
x ∉ L
qaccept
qreject
into a machine with this behavior?
qaccept
qreject
March 6, 2019
CS21 Lecture 23

11. coNP language L is in coNP iff its complement (co-L) is in NP
it is believed that NP ≠ coNP
note: P = NP implies NP = coNP
proving NP ≠ coNP would prove P ≠ NP
another major open problem…
March 6, 2019
CS21 Lecture 23

12. UNSAT = {φ : φ is an unsatisfiable 3-CNF formula}
coNP
canonical coNP-complete language:
UNSAT = {φ : φ is an unsatisfiable 3-CNF formula}
proof?
March 6, 2019
CS21 Lecture 23

13. coNP
Disjunctive Normal Form = OR of ANDs
another example
3-DNF-TAUTOLOGY = {φ : φ is a 3-DNF formula and for all x, φ(x) =1}
proof?
another example:
EQUIV-CIRCUIT = {(C1, C2) : C1 and C2 are Boolean circuits and for all x, C1(x) = C2(x)}
March 6, 2019
CS21 Lecture 23

14. Quantifier characterization of coNP
March 6, 2019
CS21 Lecture 23

15. Proof interpretation of coNP
“proof”
“short” proof
“proof verifier”
March 6, 2019
CS21 Lecture 23

16. coNP what complexity class do the following languages belong in?
COMPOSITES = {x : integer x is a composite}
PRIMES = {x : integer x is a prime number}
GRAPH-NONISOMORPHISM = {(G, H) : G and H are graphs that are not isomorphic}
EXPANSION = {(G = (V,E), 𝛼 > 0): every subset S ⊆ V of size at most |V|/2 has at least 𝛼|S| neighbors}
March 6, 2019
CS21 Lecture 23

17. coNP Picture of the way we believe things are: coNP EXP
decidable languages P NP
NP ∩ coNP
March 6, 2019
CS21 Lecture 23

18. NP ∩ coNP
Might guess NP ∩ coNP = P by analogy with RE (since RE ∩ coRE = DECIDABLE)
Not believed to be true.
A problem in NP ∩ coNP not believed to be in P:
L = {(x, k): integer x has a prime factor p < k}
(decision version of factoring)
March 6, 2019
CS21 Lecture 23

19. NP ∩ coNP
March 6, 2019
CS21 Lecture 23

20. PRIMES in NP
March 6, 2019
CS21 Lecture 23

21. Summary Picture of the way we believe things are:
(decision version of ) FACTORING
EXP
coNP
decidable languages P NP
NP ∩ coNP
March 6, 2019
CS21 Lecture 23