Determine whether or not there is a known polynomial$-$time algorithm for solving the problem $($Hamilton Cycle, Travelling salesman problem, Vertex cover, Independent Set, Finding Cliques, Subset Sum, OR Dominating Set$).$

You must justify why there is no known poly$-$time algorithm OR identify a poly$-$time procedure that solves the problem.

My teenage twins head out to the grocery store to buy a given set of items, which they will have to purchase and carry back home. There are n things on the list. The kids arrive at the store and carefully write down the weight of each to purchase. They want to be perfectly fair.... and split the carrying task equally between them. So they would like to determine if they can split the items in such a way that each child carries the exact same total weight on the way home. Note that all items must be purchased. Show that this problem is NP complete.