Chapter 34 - SHENS CLASS NOTES Chapter 34 NP-Completeness...

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SHEN’S CLASS NOTES Chapter 34 NP-Completeness 34.1 Basic Concepts After long time study, people found that some problems  can be efficiently solved, and some problems are so difficult that  only exponential time algorithms are known for them or even no  algorithm at all. In order to study the intrinsic complexity of computational  problems, we would like to classify problems into different  classes based on their difficulty levels. If a problem can be solved in time that is a polynomial  function of  n , where  n  is the input size, we say that this problem  belongs to  class P . Class  P  problems are called tractable because  they can be solved in polynomial time. A problem that requires  super-polynomial time is called intractable.    There   is   a   class   of   problems   whose   tractability   is   not  known yet. We call this class  NP-complete  (NPC). If a problem  belongs to NPC class, then it is a hard problem because no body  can provide a polynomial algorithm for it so far. However, it has  not   been   proved   yet   that   they   are   indeed   not   polynomial  solvable. 1
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SHEN’S CLASS NOTES In order to study the intrinsic complexity of problems, we  introduce the  NP class . A problem belongs to the NP class if it  can be solved by an NP algorithm in polynomial time. An NP  algorithm is an algorithm run on a powerful but hypothetical  computing model called  non-deterministic machine  model. All  NPC problems can be solved by an NP algorithm in polynomial  time. Moreover, it is proved that if any NPC problem can be  solved in the future by a (deterministic) polynomial algorithm,  then all NP problems can be solved too. This is the reason why  we use the name NP-Complete for such a problem. On the other  hand, if any NPC problem is proved to be intractable in the  future, then all NPC problems are intractable. Encodings The complexity of a problem is closely related to the input  size   which   depends   on   the   encoding   of   the   problem.   For  example, if we want to encode the number 99. Its decimal  representation, binary representation, and unary representation  are:  (99) 10 (uses two digits) (1100011)   2 (uses 7 digits) (1111…1)  1 (uses 99 digits). So, we need to say few words about the encodings.  2
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SHEN’S CLASS NOTES Two different encodings   e 1   and   e 2   are called polynomial  related for a problem I if there is a polynomial computable  functions  f  and  g  such that  f ( e 1 ( i )) =  e 2 ( i ) and  g ( e 2 ( i )) =  e 1 ( i ) for  any problem instance   i . Obviously, if encodings   e 1   and   e 2   are 
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