The input is a set S $=\{1,2,...,$ n$\}$ of n items with weights s$1,$ s$2,...,$ sn$.$ Assume, without loss of generality, that the items are ordered such that s$1$ s$2...$ sn$.$ The problem is to partition S into sets A and B such that max$($w$($A$),$ w$($B$))$ is minimized, where w$($A$)=$ X i in A si $,$ and w$($B$)=$ X j in B sj $.$ Prove the following: $$ The Partition Problem is NP$-$complete. $$ Find a PTAS for the above problem, i$.$e$.$ a $(1+)-$approximation for $>0.($Remark: Note that a $2-$approximation for the problem is easy; Define $2$L $=$ Xn i$=1$ si $=$ w$($S$).$ Then, the optimal solution will have cost Copt $$ L$.)$