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

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 = {x1, x2, ... xk }, and for some {y1, y2, … yk }  S, we have yi = 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 x1 to l be the variables and c1 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 yi if x is true else

take zi if x is false.

 If the number of true literals in cj is at most 2, take gj  If the number of true literals in cj is 1, take hj

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(n2).  Thus SUBSET-SUM is NP Complete after reducing 3SAT.

“ ”

Thank You