Question

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

Given a set of integers, we say that can be partitioned if it can be split into two sets U and V so that considering all u EU and all v € V, Eu = Ev. Let PARTITION = { <S> S can be partitioned ). Show that PARTITION IS NP-complete by reduction from SUBSET-SUM.

Answer 1

So! in the con Set one time en Venten Instance G'IVE) Vi hasone Vertex un v A Solution just need to exhibit the vertices dominating • easily verify the Solution polynomial So the problem in Np we can reduce from the Cover problem det be CV, E) bean of the vertex de fine guaph for Vertex every and one vertex for every edge e o in ce we describe El for an bedge e=(u, v) in a Now Suppose s be a vertey Cover Size ken Gil thus we can assume Scontain Vertices which form a subset of v Now et es easy that I should be a vertex Now to check cover in e need the an enpit A Solution Just to exhibet partition Since addition and multiplication take polynomial time we can easily verify easily verify a solution a a polynomial time to reduce PARTITION to this problem Recall that to partition consists of a set of position to check if they can be divided unito two has the Same Sum ef (u, v) where I is the sum of these n number Now partition into two Set U and are number n.....un we heed parts which Suppose to there were a nn equal Sem. then the same partition in

and UGU dehote Suppose it is possible to parlton the number of two parts u andu in such that su + IV II2/2 SS} and VEV the total sum of humber of Ev and su but noto this IUREV=IS EU + EV = Es with equality Eu= Ev theus Eutv
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