Three background concepts:

1

• Three background concepts
• Decision Problems output yes/no
• Nondeterministic algorithm certificate
  additional input
• Polynomial transformation one problem input to
  another type problem input in poly-time

2
1. DECISION PROBLEMS

• Decision Problems output is True/False,
  Yes/No
• Example (sorting-decision problem)
• Input a list of numbers L
• Output Answer, is L sorted?
• Complexity of Decision Problem?
• Sorting problem P lower bound O(n logn), but
• Sorting-decision problem PD O(n)

3
ADVANTAGE OF DECISION PROBLEMS

• Every problem P has a corresponding related
  decision problem PD,
• Has a lesser or equal complexity
• Logical output to work with makes it formal
• Hence, complexity theory (or, any computer
  science formal theory) uses only decision
  problems
• But, the results are valid for all types of
  problems

4
EXAMPLE DECISION PROBLEM

• Example (0-1 Knapsack-decision problem)
• Input a list of objects with (weight, profit), a
  knapsack upper bound by weight M, and a profit
  lower bound t
• Output Does there exist a ks_profit t by
  choosing some objects with ks_wt M?
• Complexity?
• Which one is lower original 01KS or 01KSD?
• 01KSD algorithm Backtrack just like before for
  01KS but may terminate as soon as a valid
  ks_profit t found, need not go over all
  branches.

5
USE OF DECISION PROBLEMS

• Algorithm A(PD ) for a decision problem PD may be
  used to solve original problem P, and vice versa
• Example of sorting, using sorting-decision
  algorithm
• Sort (input L)
• For each permutation p of input list L
• if AlgoSortingD (p) True then return p
• SortingD (input L) If any comparison within the
  algorithm Sorting(L) leads to an actual exchange
  of elements, then return False (L was not
  sorted)
• Similarly Alg(01KSD) may be used to solve 0-1KS
  Hint binary search over ks_profit bound
  variable t
• Reverse is easy ??

6
MORE CONCEPTS2. NON-DETERMINISTIC ALGORITHMS
7
NON-DETERMINISIM

• Non-deterministic version of a problem
• Presume the solution exists certificate
• Problem is simplified Check only the
  certificate,
• Is the certificate correct answer? return
  True/False
• If an algorithm can check certificate in
  polynomial time
• gt The problem is Non-deterministically
  Polynomial,
• in NP-class

8
NON-DETERMINISTIC PROBLEMSEXAMPLE

• SortingD is an ND problem
• Input (1) a list of numbers L, and (2) a
  certificate c (another list)
• Output Is c a sort of L?
• Algorithm?
• Can you find a polynomial-time algorithm?
• If your answer is yes SortingD problem is in
  NP-class
• Is SortingD also a P-class problem?

9
NON-DETERMINISIM IS NOT A MISTERY
Non-deterministic algorithm Given the
certificate, it traverses only one path from
root to a leaf bounded by depth of the tree
Deterministic algorithm Traverses all nodes of
the tree in worst case
(w88, ---) pruned
Deterministic Machine versus Non-deterministic
Machine
10
CLASSIFICATION OF PROBLEMS BY COMPLEXITY
11
COMPLEXITY OF PROBLEMS

• Polynomial algorithms are faster than
  Exponential algorithms, or more accurately
• P-algorithms has slower growth rate than that of
  E-algorithms, with respect to input size n
• O(nk) lt O(an), agt1 and k0 are constants

12
COMPLEXITY THEORY

• Each Problem has a lower bound, for Any algorithm
• What is the lower bound for Comparison Sort?
• Problems may be classified by such bounds
• An important, but unknown classification would
  be
• Polynomial-class versus Exponential-class
• Unfortunately Life is much more complicated!
• There is no E-class of problems (with exponential
  lower bounds)

13
P-CLASS PROBLEMS AND BEYOND

• P-class of problems which have asymptotic
  Polynomial-time algorithms O(nk), for k0
• There is no E-class of problems No problem S
  may be designated as in E-class just because no
  polynomial algorithm for S has been found yet!
• Even if you have an exponential algorithm O(an),
  for agt1, that does not mean a polynomial
  algorithm will also not exist!
• We are talking about provable lower bound O

14
P-CLASS PROBLEMS AND BEYOND

• Example No polynomial algorithm for
  Multi-processor Last Finish Time (MPLFT) problem
  is found yet.
• That does not mean in future a polynomial alg
  will not be found!
• Although, it is not yet in P-class, it is not
  E-class either
• No exponential lower bound has been found yet as
  well
• That is all you can say about MPLFT
• However, .continued, we use some new concepts
  .

15
P VERSUS NP

• Say, problem X is in P-class (read has a
  polynomial algorithm)
• Is X in NP-class as well?
• Read has non-deterministic Polynomial algorithm
  also?
• Read given a certificate, can you check it in
  polynomial-time?

16
P VERSUS NP

• Which one is faster?
• Polynomial Alg, or Non-det polynomial alg?
• What is more?
• Number of nodes in the tree?
• or the height of the tree?

17
P VERSUS NPP ? NP True
NP-class
P-class
18
P VERSUS NP

• Say, X is in NP-class
• Read has a non-det polynomial alg
• Is X in P-class as well?
• Read does it have a regular polynomial alg?

19
P VERSUS NPNP ? P True?
P-class
NP-class?
20
P VERSUS NP

• P ? NP is True.
• But is NP ? P also true? Meaning NP P?
• A million dollar question
• http//www.claymath.org/millennium-problems
• Jury is out there
• NP P (i.e., NP ? P and P ? NP), XOR, NP
  ? P?

21
P VERSUS NPHow would you prove NP ? P not
True?Or, that the truth is P ? NP, a proper
subset
NP-class
X
P-class
22
P VERSUS NP LEADS TO COMPLEXITY HIERARCHY

• Jury is out there
• Prove, NP ? P, and so, NP P?
• Or, show P ? NP?
• Did not happen yet!
• But, can we find a hardest problem H in NP
  class?
• And, then maybe look for a Poly-algorithm for
  that H?
• Or, find exponential lower bound for H?
• Looking for hardest needs some concepts on
  Polynomial Transformation (another new
  concept!!)

23
ANOTHER NEW CONCEPT!3. POLYNOMIAL
PROBLEM-TRANSFORMATION
24
POLYNOMIAL PROBLEM-TRANSFORMATION

• How do you formally express a problem?
• Problem (Input, Output) unambiguously expressed
• Problem instance Specific values for Input,
• Remember Output is T/F in decision problems

25
POLYNOMIAL PROBLEM-TRANSFORMATION

• Problem X can be transformed to another problem Y
• Problem Transformation an algorithm
• TXY (input of X) (input of Y)
• Such that, corresponding output of X and Y are
  same (T/F)
• TXY is an algorithm, like any other ones we have
  seen
• WAIT TO SEE EXAMPLE!

TXY
Y
X
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POLYNOMIAL PROBLEM-TRANSFORMATION

• Problem Transformation an algorithm TXY
• TXY(input of X) gt (input of Y)
• Such that, output T/F for (input of X) output
  for TXY(input X)
• If there exists a TXY,
• and its time-complexity is Polynomial,
• then X is polynomially transformed to Y

TXY
Y
X
27
SIGNIFICANCE OF POLYNOMIAL TRANSFORMATION

• If there exists a TXY, and time-complexity of it
  is Polynomial, then X is polynomially
  transformable to Y
• Suppose, Y has a polynomial algorithm Ay
• but, NO polynomial algorithm exists for X,
• YET!
• However, now you can solve X by
• Ay( TXY(X) ) ? T/F
• What is the time-complexity of this dual
  algorithm?

TXY
Y apply poly alg AY on Y
X no poly alg yet
X is True
X is False
28
A CATCH IN POLYNOMIAL TRANSFORMATION

• AY(TXY(X)) ? T/F
• What is the time-complexity of this dual
  algorithm?
• Note (Y?output(TXY(X)) is a polynomial
  function of X,
• where indicates problem size
• Ay(Y) takes polynomial-time with respect to Y
• But, Y is a polynomial of X, as said above
• Hence Ay(Y) above takes polynomial of
  polynomial-time over X,
• Or, polynomial-time with respect to X

29
SIGNIFICANCE OF POLYNOMIAL TRANSFORMATION

• Ay(TXY(X)) ? T/F
• Time-complexity of this dual algorithm is
  polynomial
• So, X is now in P-class
• Because Y was in P-class and polynomial
  transformation TXY is found
• So, X has an indirect polynomial algorithm now
• Note, this is not true in the opposite direction

30
SIGNIFICANCE OF POLYNOMIAL TRANSFORMATION

• Ay(TXY(X)) gt T/F
• If Ay is polynomial, then the time-complexity of
  this dual algorithm is also polynomial
• If Y is in P-class, so is X
• Y is at least as hard as X
• Note, opposite direction is not necessarily true
• If, source problem X is already in P-class (has
  poly alg),
• Then, polynomial transformation TXY does not say
  anything about the classification of target
  problem Y

31
COOKS THEOREM

• All NP-class problems may be expressed
  generically as a machine a Non-deterministic
  Turing Machine
• There exists a polynomial-transformation from
  this NDTM to the SAT problem Cook et al.s
  algorithm

32
COOKS THEOREM
NP-class
SAT problem
Poly-trans
P-class
33
COOKS THEOREM

• Find a polynomial algorithm for SAT,
• and win a million dollar from Clay Institute!
• Then, Cook et al.s polynomial transformation can
  be used to solve any NP-class problem in
  polynomial-time using your polynomial-SAT
  algorithm, or PNP
• Alternatively, SAT is a good candidate for
  finding an exponential lower bound
• Then, P ?NP will be proved

34
BOOLEAN SATISFIABILITY PROBLEMinPROPOSITIONAL
LOGIC(SAT)
35
SAT PROBLEM

• Input (U, V)
• U a set of n Boolean variables v1, v2,
  vn(value T, F)
• V a conjunctive normal formula (CNF) over U
• Conjunctive normal formula is conjunction of
  disjunctions
• (. v . v .) (. v . v . v .)
• Output Does there exist a satisfiable assignment
  to make C True?
• can you find values of U such that V
  becomes True?
• V expressed as a set of m Clauses C1, C2, Cm
  
• Ci is a list (disjunction) of literals (a
  variable, or its negation)
• What is the complexity of finding above output?

36
EXAMPLE SAT PROBLEM

• Input (U, V)
• U a, b, c, d
• V a, b, c, a, d, b, d, indicates
  NOT
• in CNF (a V b V c) (a V d) (b V d)
• Output Does there exist a satisfiable assignment
  to make V True?
• Answer Yes
• AT, bF, c?, d? ,
• question-mark ? means dont care or T and F
  both correct
• What is the complexity of answering the above
  question?

37
SAT is in WHICH-CLASS?

• Input (U, V)
• U a, b, c, d
• V a, b, c, a, d, b, d
• a certificate (aF, bT, cT, dT)
• Output Is the certificate a satisfiable
  assignment to make V True?
• Answer No
• C1 T, C2 T, C3 F
• V False
• What is the complexity of answering the above
  question?

38
SAT is in NP-CLASS

• Input (U, V)
• U a, b, c, d
• V a, b, c, a, d, b, d
• a certificate (aF, bT, cT, dT)
• Output Is the certificate a satisfiable
  assignment to make V True?
• Answer No
• C1 T, C2 T, C3 F makes V False
• Complexity O(nm)
• So, SAT is in NP-class
• But not in P-class, yet no one found a
  polynomial alg

39
COOK ET ALS THEOREM SAT IS NP-HARD

• Polynomial Transformation from NDTM to SAT
• SAT is NP-hard
• Either find a polynomial algorithm for SAT then
  PNP
• Or, find exponential lower bound for SAT then
  P?NP

40
COOK ET ALS THEOREM SAT IS NP-COMPLETE

• Polynomial Transformation from NDTM to SAT
• SAT is NP-hard
• NP-hard poly-trans from NDTM
• NP-complete NP-hard NP-class
• SAT is NP-class gt SAT is NP-complete

41
COOKS THEOREM
Significance IF there exists a poly-alg for
SAT, THEN PNP
or, all NP-class problems WILL have
poly-alg
42
COOK ET ALS MORE RESULT 3-SAT IS NP-HARD

• Polynomial Transformation from SAT to 3-SAT
• Significance
• IF there exists a polynomial alg for 3-SAT,
• Then, SAT would have an indirect
  poly-alg,
• So, PNP would be true,
• or, all NP-class problems
  WOULD have poly-algs
• Polynomial Transforms are Transitive
• A ? B, and B ? C, implies A?C
• Chain of poly-transformation creates NP-hard
  class

43
COOKS THEOREM LEADS TO NP-HARD CLASS
NP-hard
NP-class
3-SAT
SAT
Poly-trans
3-DM
P-class
44
COOKS THEOREM LEADS TO NP-COMPLETE CLASS
NP ? NP-hard NP-complete
NP-class
NP-complete
3-SAT
SAT
Poly-trans
3-DM
P-class
45
COOKS MORE RESULTS 3-SAT IS NP-HARD

• Chain of poly-transformation creates NP-hard
  class
• Any poly-alg found for a problem in NP-hard will
  solve PNP question (for yes)
• Overlap of NP-hard and NP-class makes NP-complete
• NP-class problem has an exponential alg. O(an),
  and
• So, is any NP-complete problem
• Branching factor of decision tree is bounded by a
  constant a

46
COOKS THEOREM LEADS TO NP-COMPLETE CLASS
NP-hard
NP-class
NP-complete
P-class
47
WHAT IS 3-SAT?

• SAT with limit on clause-size 3
• Input (U, V)
• U a, b, c, d
• V a, b, c, a, d, b, d
• This is 3-SAT ok
• We will use clause size 3, rather than 3

48
3-SAT IS NP-HARD

• Theorem There exists a poly-trans from SAT to
  3-SAT
• TXY
• Given ANY given SAT input, you can create a
  size-fixed set of clauses,
• by introducing new variables and clauses,
• such that, T/F of the ltinput SATgt is
  preserved!
• And you can do that in polynomial time with
  respect to SATs input size
• However,
• 2-SAT is in P-class David-Putnam poly algorithm
• You cannot find a polynomial-trans from SAT to
  2-SAT!

49
3-SAT IS NP-HARD

• Classify SAT clauses into 4 groups 1-clauses,
  2-clauses, 3-clauses, and p-clauses with pgt3
  literals in these clauses
• We will (1) show transformations (to the
  corresponding 3-clauses) for each of these 4
  types of clauses
• (2) prove that the transformations are correct,
  i.e., truth preserving and then
• (3) we will discuss the polynomial-time nature of
  the aggregate-transformation at the end

50
3-SAT IS NP-HARD
For each clause do
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause u u, z1, z2, u, z1, z2 u, z1, z2 u, z1, z2 New variables for each such 1-clause z1, z2,



WHY?
u is a literal
Remember 3 clauses are created, and 2 NEW
variables added in target 3-SAT for EACH
1-clause in the source SAT
51
3-SAT IS NP-HARD
For each clause do
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause u u, z1, z2, u, z1, z2 u, z1, z2 u, z1, z2 z1, z2,
For each 2-clause u1, u2 u1, u2, z1 u1, u2, z1 New variable for each such clause z1


u1, u2 are literals
Target 2 clauses created, and 1 new variable
added Source for each 2-clause
52
3-SAT IS NP-HARD
For each clause do
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause u u, z1, z2, u, z1, z2 u, z1, z2 u, z1, z2 z1, z2,
For each 2-clause u1, u2 u1, u2, z1 u1, u2, z1 z1
For each 3-clause

? clauses created, and ? new variable added
for each 3-clause
53
3-SAT IS NP-HARD
For each clause do
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause u u, z1, z2, u, z1, z2 u, z1, z2 u, z1, z2 z1, z2,
For each 2-clause u1, u2 u1, u2, z1 u1, u2, z1 z1
For each 3-clause u1, u2, u3 u1, u2, u3 No new variable for any 3-clause
For each p-clause (pgt3) u1, uk-1,uk,uk1, up u1, u2, z1, z1, u3, z2, z2, u4, z3, , zk-3,uk-1,zk-2, zk-2, uk, zk-1,zk-1,uk1,zk, , zp-3, up-1, up New variable for each such clause z1, z2, , zp-3
Target (p-2) new clauses created, and (p-3) new
variable added Source for each 3-clause
54
3-SAT IS NP-HARD
For each clause do
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause u u, z1, z2, u, z1, z2 u, z1, z2 u, z1, z2 z1, z2,
For each 2-clause u1, u2 u1, u2, z1 u1, u2, z1 z1
For each 3-clause u1, u2, u3 u1, u2, u3 -
For each p-clause (pgt3) u1, uk-1,uk,uk1, up u1, u2, z1, z1, u3, z2, z2, u4, z3, , zk-3,uk-1,zk-2, zk-2, uk, zk-1,zk-1,uk1,zk, , zp-3, up-1, up z1, z2, , zp-3
Complexity? Input SAT problem size n variables
in U, and m clauses in V k1 of 1-clauses, k2 of
2-clauses, k3 of 3-clauses, kp for each p-clauses
m k1k2k3 ?p kp
55
3-SAT IS NP-HARD
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause u u, z1, z2, u, z1, z2 u, z1, z2 u, z1, z2 z1, z2,
For each 2-clause u1, u2 u1, u2, z1 u1, u2, z1 z1
For each 3-clause u1, u2, u3 u1, u2, u3 -
For each p-clause (pgt3) u1, uk-1,uk,uk1, up u1, u2, z1, z1, u3, z2, z2, u4, z3, , zk-3,uk-1,zk-2, zk-2, uk, zk-1,zk-1,uk1,zk, , zp-3, up-1, up z1, z2, , zp-3
Complexity Input SAT problem size n variables
in U, and m clauses in V where m k1k2k3 ?p
kp k1 of 1-clauses, k2 of 2-clauses, k3 of
3-clauses, kp for each p-clauses Output of TXY
Nunmber of 3-variables n 2k1 k2 ?p
(p-3)kp Number of 3-clauses 4k1 2k2 k3 ?p
(p-2)kp Time-complexity of transformation One
step per variable or clause creation Add above
two, a
polynomial with respect to n m
56
Is 3-SAT also NP-complete?

• Trivial, SAT is in NP-class, so is 3-SAT
• 3-SAT is a set of 3-clauses over a set of Boolean
  variables
• Certificate is a given value for each variable.
• In O(nm) time the clauses may be checked for
  satisfiability
• 3-SAT is NP
• We already proved 3-SAT is NP-hard
• gt 3-SAT is NP-complete

57
EXAMPLE SAT TO 3-SAT
Input SAT problem U a, b, c, d, e, f, g, U
n 7 V (a, b), (a, c, d), (b, c, d, e,
f), (g), (a, c, b, e, f, g), V m 5
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause u u, z1, z2, u, z1, z2 u, z1, z2 u, z1, z2 z1, z2,
For each 2-clause u1, u2 u1, u2, z1 u1, u2, z1 z1
For each 3-clause u1, u2, u3 u1, u2, u3 -
For each p-clause (pgt3) u1, uk-1,uk,uk1, up u1, u2, z1, z1, u3, z2, z2, u4, z3, , zk-3,uk-1,zk-2, zk-2, uk, zk-1,zk-1,uk1,zk, , zp-3, up-1, up z1, z2, , zp-3
Output 3-SAT problem U3 a, b, c, d, e, f, g,
V3
58
EXAMPLE SAT TO 3-SAT
Input SAT problem U a, b, c, d, e, f, g, U
n 7 V (a, b), (a, c, d), (b, c, d, e,
f), (g), (a, c, b, e, f, g), V m
5 1-clauses (g)
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause (g) g, z1, z2, g, z1, z2 g, z1, z2 g, z1, z2 z1, z2,
For each 2-clause
For each 3-clause
For each p-clause (pgt3)
Output 3-SAT problem U3 a, b, c, d, e, f, g,
z1, z2, V3 (g, z1, z2), (g, z1, z2),
(g, z1, z2), (g, z1, z2),
59
EXAMPLE SAT TO 3-SAT
Input SAT problem U a, b, c, d, e, f, g, U
n 7 V (a, b), (a, c, d), (b, c, d, e,
f), (g), (a, c, b, e, f, g), V m
5 2-clauses (a, b)
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause (g) g, z1, z2, g, z1, z2 g, z1, z2 g, z1, z2 z1, z2,
For each 2-clause (a, b) (a, b, z3), (a, b, z3) z3
For each 3-clause
For each p-clause (pgt3)
Output 3-SAT problem U3 a, b, c, d, e, f, g,
z1, z2, z3, V3 (g, z1, z2), (g, z1,
z2), (g, z1, z2), (g, z1, z2), (a, b, z3),
(a, b, z3)
60
EXAMPLE SAT TO 3-SAT
Input SAT problem U a, b, c, d, e, f, g, U
n 7 V (a, b), (a, c, d), (b, c, d, e,
f), (g), (a, c, b, e, f, g), V m
5 3-clauses (a, c, d)
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause (g) g, z1, z2, g, z1, z2 g, z1, z2 g, z1, z2 z1, z2,
For each 2-clause (a, b) (a, b, z3), (a, b, z3) z3
For each 3-clause (a, c, d) (a, c, d)
For each p-clause (pgt3)
Output 3-SAT problem U3 a, b, c, d, e, f, g,
z1, z2, z3, V3 (g, z1, z2), (g, z1,
z2), (g, z1, z2), (g, z1, z2), (a, b, z3),
(a, b, z3), (a, c, d),
61
EXAMPLE SAT TO 3-SAT
Input SAT problem U a, b, c, d, e, f, g, U
n 7 V (a, b), (a, c, d), (b, c, d, e,
f), (g), (a, c, b, e, f, g), V m
5 5-clauses (b, c, d, e, f)
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause (g) g, z1, z2, g, z1, z2 g, z1, z2 g, z1, z2 z1, z2,
For each 2-clause (a, b) (a, b, z3), (a, b, z3) z3
For each 3-clause (a, c, d) (a, c, d)
For each p-clause (pgt3) (b, c, d, e, f) (b, c, z4), (z4, d, z5), (z5, e, f) z4, z5
Output 3-SAT problem U3 a, b, c, d, e, f, g,
z1, z2, z3, z4, z5 V3 (g, z1, z2), (g,
z1, z2), (g, z1, z2), (g, z1, z2), (a, b,
z3), (a, b, z3), (a, c, d), (b, c, z4),
(z4, d, z5), (z5, e, f),
62
EXAMPLE SAT TO 3-SAT
Input SAT problem U a, b, c, d, e, f, g, U
n 7 V (a, b), (a, c, d), (b, c, d, e,
f), (g), (a, c, b, e, f, g), V m
5 6-clauses (a, c, b, e, f, g)
Input SAT clause-type Input SAT-clause Output 3SAT clause Output 3-SAT new variables
For each 1-clause (g) g, z1, z2, g, z1, z2 g, z1, z2 g, z1, z2 z1, z2,
For each 2-clause (a, b) (a, b, z3), (a, b, z3) z3
For each 3-clause (a, c, d) (a, c, d)
For each p-clause (pgt3) (b, c, d, e, f) (b, c, z4), (z4, d, z5), (z5, e, f) z4, z5
(a, c, b, e, f, g) (a, c, z6), (z6, b, z7), (z7, e, z8), (z8, f, g) z6, z7, z8
Output 3-SAT problem U3 a, b, c, d, e, f, g,
z1, z2, z3, z4, z5, z6, z7, z8, U3 7
12 11 10 (2 3) 16 V3 (g, z1,
z2), (g, z1, z2), (g, z1, z2), (g, z1,
z2), (a, b, z3), (a, b, z3), (a, c, d), (b,
c, z4), (z4, d, z5), (z5, e, f), (a, c, z6),
(z6, b, z7), (z7, e, z8), (z8, f, g),
V314 1211 (34) 14
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SUMMARY ON COMPLEXITY

P-class problem Polynomial algorithm exists
NP-class problem Certificate checking in polynomial. Must have exponential algorithm.
P in NP? Yes. Poly-alg for original indicates poly-alg for certificate checking.
NP in P? Not known.
Polynomial transformation. X to Y poly-trans means X can be solved by solving Y. If Y is in P, then so is X.
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SUMMARY ON COMPLEXITY

NP-hard All NP-class problems can be poly-transformed to any NP-hard problem
NP-complete NP-hard and NP-class
When to prove a problem Y is NP-complete? Intuition tells so.
How to prove Y is NP-complete? (a) Find a poly-trans from an NP-hard problem X to Y - (1) truth-preserving, and (2) poly-time. (b) Prove Y is NP-class.
Any other way to prove Y is NP-hard? Yes, many. E.g. prove a restricted version of Y is an NP-hard problem. Not the other way round 2-SAT is P-class. https//en.wikipedia.org/wiki/DPLL_algorithm is poly-time for 2-SAT
A poly-alg is found for an NP-hard problem (by a nut/ a genius!). What would that mean? Not truth preserving, or (2) not poly-trans, or (3) source problem (X) was not really NP-hard, or, Wrong proof. Otherwise, (3) NPP is just being proved!
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SUMMARY ON COMPLEXITY

Examples of NP-complete problems 1. 0-1 Knapsack decision problem
2. Multi-processor Last-time-finish scheduling problem
3. Hamiltonian Circuit decision problem Given a graph G, does there exists a cycle involving each node once and only once? Does there exist a cycle in G involving each arc once and only once? Euler circuit problem P-class
4. 3 dimensional matching (3DM) Given three equal size sets X, Y, Z of three types, does there exists a perfect matching? Matching set A, a set of 3 elements (marriage), one from each of X, Y, and Z. Perfect matching (subset of A) each element of X, Y, Z appears in one and only one marriage in A. 2DM is P-class, but 3DM is NP-complete
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HOW TO DEAL WITH NP-COMPLETE PROBLEM
Y is proved NP-complete
Restrict Y to something Z, then see if poly-alg may be developed for Z SAT to 2-SAT
Approximate solution for Y may be acceptable e.g. Multi-processor LFT scheduling problem
A pseudo-poly alg may work most of the time for Y 0-1 Knapsack Dynamic Programming
Apply pruning in Branch-and-bound exponential-time alg for Y We use such smart exp algorithms for most NP-complete problems. Do not care about the worst-case complexity, that is still exponential! NP-complete problems have additional inherent structures!

How about parallel computing? Does not solve an NP-comp problem in poly-time
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MORE ON COMPLEXITY ISSUES

Co-NP Complementary to NP-class. Overlaps NP
Co-NP complete Poly trans from all Co-NP, and inside Co-NP
P-space Remember, space complexity is less than or equal to time complexity!
P-space complete Harder than NP-complete.
Infinite hierarchy
Quantum computing-hierarchy Inherently Non-deterministic computing model, but has its own hierarchy. Some NP-complete problems (TSP) do get solved in poly-time (non-deterministically)
Undecidable problem Definition No halting Turing machine exists, i.e. No algorithm exists in general. This is NOT about complexity!