Let G = (V, E) be a graph with rational-valued edge weights (ce : e ? E). (The edge weights may include negative numbers.) For any S ? V such that ? 6= S 6= V , let ?(S) = {e ? E : e has exactly one end in S}. Formulate an integer programming model to find ? 6= S 6= V that minimizes P(ce : e ? ?(S)). The number of variables and constraints in your model must be at most a|V | + b|E| + c for some constants a, b, c (the values of a, b, c cannot depend on the values of |V | and |E|).

2. (5 marks) Let G = (V. E) be a graph with rational-valued edge weights (Ce : e E E). (The edge weights may include negative numbers.) For any S C V such that 0 # S # V, let 6(S) = fe E E : e has exactly one end in S}. Formulate an integer- programming model to find 0 / S / V that minimizes C(Ce : e E .(S)). The number of variables and constraints in your model must be at most a V| + b E| + c for some constants a, b, c (the values of a, b, c cannot depend on the values of | V | and | E))