NP-Completeness  For convenience, the theory of NP - Completeness is designed for decision problems (i.e. whose solution is either yes or no).  Abstractly, a decision problem  consists simply of a set D   of instances and a subset Y   D   We describe a decision problem by specifying a generic instance and stating a yes - no question  Examples: subgraph isomorphism and TSP

Decision Problems  A decision problem can be obtained from an optimization problem by adding a bound  As long as the cost function is relatively easy to compute, the decision problem can be no harder than the optimization problem.  The reason we restrict ourselves to decision problems is that they have a natural correspondence to formal languages.

Formal languages  For any finite set of symbols , we denote by  * the set of all finite strings of symbols from . For example, if  = {0,1}, then  * consists of , 0,1, 01,10,00,11,…….  If L is a subset of  *, we say L is a language over the alphabet .  An encoding scheme is a mechanism for describing each instance of a problem  by an appropriate string of symbols over some fixed alphabet .

Problem Encodings  Thus a problem  and encoding scheme e partition  * into three classes : those that are encoding of "no", those that are encoding of "yes", and those that are not a valid encoding at all.  We define [ ,e] = { x   * :  is the alphabet of e and x is the encoding under e of an instance I  Y  }  We will say that if a result holds for L[ ,e] then it holds for problem  under encoding scheme e

Problem Encodings  Most properties that we will consider will be essentially encoding independent so long as we restrict ourselves to reasonable encoding schemes  That is, if e and e are both reasonable encoding schemes, then a property holds either for both L[ ,e] and L[ , e ] or neither.  This will allow us to say that the property holds for  without specifying e.

Reasonable Encoding Scheme  conciseness  no unnecessary padding  binary representation of numbers  Decodability  Given any particular component of a given instance, one should be able to specify a polynomial-time algorithm that is capable of extracting its value from an encoded instance

Turing Machine  Deterministic one_tape Turning Machine (DTM)  Consists : finite state control, read_write head, and a tape (2_way infinite)  A program for a DTM specifies  A finite set  of tape symbols which includes    of i/p symbols and a distinct blank symbol b   -   A finite set Q of states which includes q 0, q y, q N  : (Q - { q y, q N } )    Q    { -1,1}

Turing Machine Operation  i/p is a string x   *.  x is placed in square 1…|x|. All others contain blank.  Program starts in state q 0 at sequence 1.  If current state is q y or q N then computation has ended and the answer is "yes" or "no".

Complexity Classes  P = { L :  polynomial time DTM m for which L= L m }  NP = { L :  polynomial time NDTM m for which L= L m }  NP can also be defined in terms of polynomial-time verifiability

Verifiability  Take TSP. Suppose someone claimed that the answer for some instance is "yes".  We could, if we were skeptical, ask this person to prove their claim by providing us with a tour.  It would then be a simple matter to verify the truth or falsity of their claim.

Verifiability  It is this notation of polynomial time verification that the class NP is intended to isolate. Note that polynomial verifiability is different from solvability

Polynomial Reducability  A language L is NP -complete if L  NP and for all other language L  NP, L  L  If any single NP complete problem can be solved in polynomial time, then all problems in NP can be solved in polynomial time  If any problem in NP -complete is intractable, then all problems in NP -complete are intractable  Once we have even a single NP -complete problem, it becomes easier to prove NP - completeness

co- NP co- NP ={L | complement (L)  NP } Is ( NP  co- NP ) - P non-empty?

Proving NP -Completeness A problem  is NP -complete if it satisfies two properties: –If  belongs to NP [this condition is sometimes called being “ NP -easy”] –If every problem in NP can be reduced to  in polynomial time [this condition is called being “ NP -hard”]