Consider the following problems:

Let $ G=(V,E)$ be an undirected graph and $ k$ be a natural number. k-Independent-Set asks if there exists a subset $ C\subseteq V$ of size $ k$ of nodes from $ G$ such that all nodes from $ G$ are independent: $ \forall u,v \in V . (u,v)\not\in E$ .

Remark: Let $ G$ be a social network where each edge $ (u,v)\in E$ models friendship between participants $ u$ and $ v$ . k-independent-set asks if there exist a group of size $ k$ such that no two members of it are friends.

Let $ a_1, a_2, \ldots, a_n, b$ be natural numbers. Subset Sum asks if there exist a subset of $ a_1, a_2, \ldots, a_n$ whose sum equals $ b$ .

Remark: Imagine $ a_1, a_2, \ldots, a_n$ to be weights of various items identified as $ 1,2, \ldots, n$ and that $ b$ is the capacity of a rucksack. Subset sum asks if you can pick a combination of items such that the rucksack can be completely full.

Let $ a_1, a_2, \ldots, a_n$ be natural numbers. Partition asks if there exists a partitioning of $ {a_1, \ldots, a_n}$ into $ P_1,P_2$ such that the sum of elements from one element of the partition equals that of the other.

• how is a partition formally defined?

Remark: Imagine you have items each having a certain value $ a_1, a_2, \ldots, a_n$ . Partition asks you to split those items between two people such that each one receives equal value.

1. Show that