Theorem 7.56 SUBSET-SUM is NP Complete ANSHUMAN MOHANTY SUBSET-SUM - PowerPoint PPT Presentation

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Theorem 7.56 SUBSET-SUM is NP Complete ANSHUMAN MOHANTY SUBSET-SUM Problem Consider a set of numbers S. and a target number t. We have to determine of the exists a subset of S such that the summation of the numbers present in the subset is

Theorem 7.56 SUBSET-SUM is NP Complete ANSHUMAN MOHANTY
SUBSET-SUM Problem  Consider a set of numbers S. and a target number t. We have to determine of the exists a subset of S such that the summation of the numbers present in the subset is equal to t.  The subset is considered a multi-set i.e. repetitions are allowed.  Thus we have, SUBSET-SUM = {  S,t  | S = {x 1 , x 2 , ... x k }, and for some {y 1 , y 2 , … y k }  S, we have  y i = t)
NP and NP Complete  NP is the class of languages that have polynomial time verifiers. i.e. we can verify if a given solution is true or not for a given problem in polynomial time.  A problem P is said to be NP Complete if the it belongs to NP and if there exists a problem in NP that is reducible to P.
SUBSET-SUM is NP
SUBSET-SUM is NP Complete  The idea of the proof is to reduce 3SAT NP Complete problem to SUBSET-SUM.  We create a 3 cnf formula  to construct an instance of SUBSET- SUM which contains a subcollection whose summation is target t.  Using variables and clauses we find structure to the SUBSET-SUM  Each variable x is represented by y and z which establishes the truth value of x in the subcollection.
Proof  Let x 1 to l be the variables and c 1 to k be the clauses.  We construct a table with l+k columns and 2(l+k) rows.  For every y, z in the i-th row, set the i-th column value as 1 and the rest be 0.  For every y, z in the j-th row, set y = 1 for the j-th column if x is true else set z as true. Everything else is to be set as 0.  For every g, h in the (l+j)-th row, set the j-th column as 1 and set everything else as 0.  Set the first l columns of final row as 1 and the remaining k columns as 3.
Subset exists if formula is satisfiable  Considering a sample Boolean expression, take y i if x is true else take z i if x is false.  If the number of true literals in c j is at most 2, take g j  If the number of true literals in c j is 1, take h j
Example
Conclusion  We can conclude that the reduction can take place in polynomial time.  The size of the table is finite, roughly (l+k) 2 and that each entry can be easily calculated for any   The time complexity is O(n 2 ).  Thus SUBSET-SUM is NP Complete after reducing 3SAT.
“ Thank You ”

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