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 Î U and all v Î V, Su = Sv. Let PARTITION = { | S can be partitioned }.

Show that PARTITION is NP-complete by reduction from SUBSET-SUM

Answer 1

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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 Î U and all v Î V, Su = Sv. Let PARTITION = { | S can be partitioned }.

Show that PARTITION is NP-complete by reduction from SUBSET-SUM

b) by even & show that PARTITION ES Nlp- complete by reeduction freon SUBSET-SOM 1) There is a non-deferemeni polynomial-tuye algore the that solves A is fe.Np. "") Any Np- compute probley B can be reduced to of. 4) The reeduction of B to A works in polynomery teme. iv). The Original probley. A has a sou of and Only Of B has a soketºon. We know", show that SET-PARTITION os Np-compute to see that 8+T - PARTITION can be verified en polynomial time; a solution It is easy

1 +20 Given partitions Uv just sum the two and verify that theire sum equals each other, which is obuch delly a polynoucas the teme verification. (Because, lummation is a polynoyicy operation and we barce performing at most Is1 many greumation) The core of the proof es en reeaducing substtt. to PARTITION, To that end geven eets and a value to one lubcet cumy query we fory: a new set slasi s vf x-27?, where x = 2 a 3 To see that there is a reduction. there erists some pes seech that t-Ex 1 (7) Assupe nep then we would have that x-t = EX xep us x-2t} -t = Ta XES' I(PU{ x=2+7) and we would have that pusx-26 f and e'l (pU { x=2+] 4204 a paretition of sl. (*) suppose that Such that there & a pare te teon Piip of Ex Ex nepi REP

Notice that thee Endvere's a natural pare to ton P, and P2 ofs, such that wlog we have that x-2t + Exs Zx XEPI xepi 3) X-2t + E + [xo=Extła zx xep, XGP2 nep, NEP os-2t +222-X NEPI we 22 Pu{x-2+} Znot nep, Hence, from a solution t-Ex eam nep from partetion Pi-pu. P2 - PU S X-2 conversely from a parte teug Pis P' Solcetion t29 !!{x-24} and there force the happing f: (sit) y el is a (Because eest) es the language feet LUBLETSU = f(sit) is the language / set (pare té te'on). PARTEOTON) can from xEpi reeduction ETSUM ای د

was And it is cleare to see that the transformation done in polynomial teme. Hences PARTINEON es Np-compute.