1. CSci 4011
INHERENT LIMITATIONS OF COMPUTER PROGRAMS

2. HARDEST PROBLEMS IN NP Definition: A language B is NP-complete if:
Theorem: A language B is NP-complete if:
1. B  NP
2. A is NP-complete and A ≤P B
2. Every A in NP is poly-time reducible to B
(i.e. B is NP-hard)
If B is NP-Complete and P  NP, then
There is no fast algorithm for B.

3. REDUCTION STRATEGIES A reduction by restriction shows that the source
problem is a special case of the target problem.
For example, 3SAT ≤P CNF-SAT because every
satisfiable 3CNF is also a satisfiable CNF.
Example. Prove 3SAT ≤P 4SAT by restriction.
A 3CNF can be converted to an equivalent 4CNF
by repeating one literal in each clause.

4. REDUCTION STRATEGIES A reduction from A to B by local replacement
shows how to “translate” between “units” of A
and “units” of B.
Example. vertex cover “units” are vertices and
edges; set cover “units” are elements and sets.
Example. CIRCUIT-SAT units are gates, CNFSAT units are clauses, 3SAT units are 3-literal clauses.

5. GRADUATION A transcript is a set of course numbers a student has taken
A major consists of:
Pairs: exactly one of which must be taken
Lists: at least one course of which must be taken
GRADUATION = {〈T,M〉 | a subset of T satisfies M}
For example:
T = {1901A, 1902B, 1902A, 2011, 4041A, 4061, 4211}
M = [1901A,1901B], [1902A,1902B]
(4011,4041A,4041B), (4211,4707), (4061)

6. GRADUATION ∈ NP:
The subset is a proof that (T,M) ∈ GRADUATION.
3SAT ≤P GRADUATION:
T = {101, 102,
201, 202,
301, 302}
(x1 ⋁ x2 ⋁ ¬x3) ∧
(¬x1 ⋁ x2 ⋁ x2) ∧
(¬x2 ⋁ x3 ⋁ x1)
M = [101, 102],
[201, 202],
[301, 302]
(101,201,302),
(101,201,201),
(101,202,301)

7. GRADUATION ∈ NP:
The subset is a proof that (T,M) ∈ GRADUATION.
3SAT ≤P GRADUATION:
Let  = C1 ∧ C2∧ … ∧Cm have variables x1…xk
For each xi:
add classes i01 and i02 to T.
add pair (i01,i02) to M.
For each Cj, we add a triple to M:
if xi is a literal in Cj, the triple includes i01.
if ¬xi is a literal in Cj, the triple includes i02.

8. UHAMPATH No HAM PATH Undirected HAM PATH A B A B C D C D E F E F
UHAMPATH = {〈G,s,t〉 | G is an undirected graph
with a Hamiltonian path from s to t}

9. HAMPATH ≤P UHAMPATH
Amid A B
Aout
Ain C D E F

10. HAMPATH ≤P UHAMPATH A B C D E F

11. HAMPATH ·P UHAMPATH f(G,s,t) = (G’,s’,t’) where: For each node u  G:
add nodes uin, uout, umid to G’.
add edges {uin,umid} and {umid,uout} to G’
For each edge (u,v)G, add {uout,vin} to G’
s’ = sin, t’ = tout.
If 〈G,s,t〉 HAMPATH, then 〈G’,s’,t’〉UHAMPATH:
(s,u,v,..,t) ! (sin,smid,sout,uin,umid,uout,vin,vmid,vout,…tout)

12. If v2  smid, then no vi=smid. So v2=smid, v3=sout
HAMPATH ·P UHAMPATH
If 〈G’,s’,t‘〉 2 UHAMPATH, then 〈G,s,t〉2 HAMPATH:
Let (sin=v1,v2,…,v3n=tout) be the undirected path.
Claim: for all i≥0, there exists u  G so that:
v3i+1=uin, v3i+2=umid, v3i+3=uout.
i=0:
If v2  smid, then no vi=smid. So v2=smid, v3=sout
Induction: if v3i = u’out, then v3i+1=uin, so v3i+2=umid.
The directed Hamiltonian path is (u1, u2, …, un)

13. 3SAT P SUBSET-SUM We transform a 3-cnf formula  into 〈y1…yn, t〉:
  3SAT  〈y1…yn,t〉  SUBSET-SUM
The transformation can be done in time polynomial in the length of 

14. x2 x1 x1 (x1 ⋁ x2 ⋁ x2) ∧ (¬x1⋁ x2 ⋁ x2) ∧ (x1 ⋁ ¬x2 ⋁ x2) x2
Each variable and each clause result in two yi’s. Each yi will have a digit for each clause and variable. x2 x1 C3 C2 C1 y1 1 y2 y3 y4 y5 y6 y7 y8 y9
y10 t 3 x1
(x1 ⋁ x2 ⋁ x2) ∧
(¬x1⋁ x2 ⋁ x2) ∧
(x1 ⋁ ¬x2 ⋁ x2) x2 C3 C2 C1

15. REDUCTION STRATEGIES A reduction by component design constructs
gadgets that simulate variables and clauses.
In particular a reduction must provide:
a gadget to force the solution to the target
instance to choose either x or ¬x
a gadget to force the solution to the target
instance to choose one literal from each clause
a gadget to force the solution to the target
instance to satisfy every clause

16. K-CLIQUES b a e c d f g

17. 3SAT P CLIQUE We transform a 3-cnf formula  into 〈G,k〉 such that
  3SAT  〈G,k〉  CLIQUE
The transformation can be done in time polynomial in the length of 

18. (x1  x1  x2)  (x1  x2  x2)  (x1  x2  x2)
k = #clauses

19. (x1  x1  x1)  (x1  x1  x2)  (x2  x2  x2)  (x2  x2  x1)

20. 3SAT P CLIQUE We transform a 3-cnf formula  into 〈G,k〉 such that
  3SAT  〈G,k〉  CLIQUE
If  has k clauses, we create a graph with k clusters of 3 nodes each. Each cluster corresponds to a clause. Each node in a cluster is labeled with a literal from the clause.
We do not connect any nodes in the same cluster
We connect nodes in different clusters whenever they are not contradictory

21. 3SAT P CLIQUE We transform a 3-cnf formula  into 〈G,k〉 such that
  3SAT  〈G,k〉  CLIQUE
 is satisfiable iff there is an assignment that
makes at least one literal true in each clause.
The nodes corresponding to these k true literals
do not contradict each other, so they form a k-
clique in G.
A k-clique in G must have one node from each
cluster. Setting the corresponding literals to true
will satisfy 

22. VERTEX COVER Theorem: VERTEX-COVER is NP-Completeb a e c d b a e c d
Theorem: VERTEX-COVER is NP-Complete
(1) VERTEX-COVER  NP
(2) CLIQUE ≤P IND-SET ≤P VERTEX-COVER

23. VERTEX-COVER = { 〈G,k〉 | G is an undirected graph with a k-node vertex cover }
Theorem: VERTEX-COVER is NP-Complete
(1) VERTEX-COVER  NP
(2) 3SAT P VERTEX-COVER

24. 3SAT P VERTEX-COVER
We transform a 3-cnf formula  into 〈G,k〉 such that
  3SAT  〈G,k〉  VERTEX-COVER
The transformation can be done in time polynomial in the length of 

25. (x1  x1  x2)  (x1  x2  x2)  (x1  x2  x2)
k = 2(#clauses) + (#variables)

26. HAMILTONIAN PATHS b a e c d f h i g

27. HAMPATH = { 〈G,s,t〉 | G is a graph with a Hamiltonian path from s to t }
Theorem: HAMPATH is NP-Complete
(1) HAMPATH  NP
(2) 3SAT P HAMPATH

28. 3SAT P HAMPATH We transform a 3-cnf formula  into 〈G,s,t〉 so that
  3SAT  〈G,s,t〉  HAMPATH
The transformation can be done in time polynomial in the length of 

29. (x1 Ç x2 Ç x2) Æ (:x1 Ç x2 Ç x2) s x1 x1 x2 x2 t

30. (x1 Ç x2 Ç x2) Æ (:x1 Ç x2 Ç x2) s x1 c1
x11
x12
x21
x22 x1 x2 x2 t

31. (x1 Ç x2 Ç x2) Æ (:x1 Ç x2 Ç x2) s x1 x11 x12 x14 x15 x1 c1 c2 x2 x21t

32. 3-COLORING b a e c d b a e c d a a d d b b c

33. 3SAT P 3COLOR We transform a 3-cnf formula  into a graph G so
  3SAT  G  3COLOR
The transformation can be done in time polynomial in the length of 

34. (x1 Ç x2 Ç x2) ? x1 x2 F T

35. (x1 Ç x2 Ç x2) ? x1 x1 F T x2 x2
c11
c12
c13
c14
c15
c16

36. (x1 Ç x2 Ç x2) Æ (x2 Ç :x1 Ç :x1) ? x1 x1 F T x2 x2 c11 c12 c13 c11

37. (x1 Ç x2 Ç x2) Æ (x2 Ç :x1 Ç :x1) ? x1 x1 F T x2 x2 c11 c12 c13 c11

38. MORE GADGETS

39. MORE GADGETS