The discrete optimization problems with interval objective function on graphs and hypergraphs and the interval greedy algorithm
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We consider the discrete optimization problems with interval objective function on graphs and hypergraphs. For the problems, we need to find either a strong optimal solution or a set of all possible weak solutions. A strong solution of the problem is a solution that is optimal for all possible values of the objective function's coefficients that belong to predefined intervals. A weak solution is a solution that is optimal for some of the possible values of the coefficients. We characterize the strong solutions for considered problems. We give a generalization of the greedy algorithm for the case of interval objective function. For the discrete optimization problems we consider, the algorithm gives a set of all possible greedy solutions and the set of all possible values of the objective function for the solutions. For a given probability distribution that is defined on coefficients' intervals, we compute probabilities of the weak solutions, expected values of the objective function for them, etc.
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