Question

Given a set S of integers, we say that S can be partitioned if it can be split into two sets U and V so that considering all u  U and all v  V, u = v. Let PARTITION = { | S can be partitioned }. a. (5) Show that PARTITION  NP by writing either a verifier or an NDTM. b. (15) Show that PARTITION is NP-complete by reduction from SUBSET-SUM.

Answer 1

ANSWER :

< vs & Cheek that and unv= 0 a) PARTITION ONP - 7 Hence is a polynoucas - Alue Verci firem fore PARTITION: On input ( juiv): (UUU - ♡ (where un uncon and Az interesection) & check the seep of elements in u equals the sum of elements en v. * 17 so accept : Otheruese , teefect. By given conditoon Eu- Ev, tur ud vet Hences PARTIOTION END. Ore the pare to tion problem Os on the class Np. b) by even & Show that . PARTITION és Np-complete by reduction from SUBSET-SUM 1) There is a non-deteremene polynomial-teme algorithe that solves A is fenp. "") Any Np-compude probley B can be reduced to f. tu) The reeduction of B to A works in polynomery temp. iv) The Original probley. A has a only of B has a someteo. We toow, show that SET – PARTITION 9 Np-compute easy SET - PARTITION can be verified en polynomial time; So letecom of and Ito to see that

) Goven, partitions U, V jest sum the two and verify that their sem equals each other, which is obver dely a polynomear the teme verefeation. (Because , summation & a polynoycay operation and we Barice performing at most 1 sl many seamation) core of the proof is en reedeling suBee7+ t som to PARTITIONS To that end geven sets and a value to the subet sum query. we forey" a new set slas The su{x-24} a where x = E a To see that this is a reduction. (7) Assume theree erists some pes such that t-Ex nep then we would have that x-t = 2x xep of x-2t} x-t Z XES' and we would have that pusx-2tz and s'I(PU{ x =2+3) from a paretetion of sl. (E) suppose that such that there is a pare te toon Piip of En Ex rep REPL

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And It is cleare to see that the transformation was done in polynomial teme. TITRON Es Np-com compute. Hence, PART تر0 کی

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