Consider an arbitrary undirected graph where each node $i$ has an arbitrary cost $c_i>0$ and each edge has a weight of 1. The objective is to select a set of edges so that the total weight of the selected edges minus the total cost of the nodes incident to the selected edges is maximized. For example, if edge $(1,2)$ and $(2,3)$ are selected, then the objective value is 1+1-$c_1$-$c_2$-$c_3$. I strongly feel that this problem is NP complete (or NP hard?), but I do not know how to prove or disprove it. It seems hard to reduce from a classic NP complete problem like set cover or independent set problem. Can anyone provide any hint?
2026-03-26 16:03:46.1774541026
A simple graph problem that seems NP complete
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