Partition Problem

Problem Definition

Given integers $v_1,\cdots ,v_n$, is there a subset $$\begin{gathered} S\subseteq \\ 1,\cdots ,n \end{gathered}$$ such that ${\sum }_{i\in S}{v}_i={\sum }_{i\notin S}{v}_i$.

Prove Partition problem is NP-complete

1. Prove it is NP.
2. Choose Subset Sum Problem.
3. Prove that Subset Sum Problem ${\le }_P$ Partition problem. Given Subset Sum instance $w_1,\cdots ,w_n$ and an integer $W$, set $v_i=w_i$, and $W=\frac{1}{2}{\sum }_{i\in [n]}{v}_i$.