Posimodular Function Optimization

Given a posimodular function f : 2 → R on a finite set V , we consider the problem of finding a nonempty subsetX of V that minimizes f(X). Posimodular functions often arise in combinatorial optimization such as undirected cut functions. In this paper, we show that any algorithm for the problem requires Ω(2 n 7.54 ) oracle calls to f , where n = |V |. It contrasts to the fact that the submodular function minimization, which is another generalization of cut functions, is polynomially solvable. When the range of a given posimodular function is restricted to be D = {0, 1, . . . , d} for some nonnegative integer d, we show that Ω(2 d 15.08 ) oracle calls are necessary, while we propose an O(nTf + n )-time algorithm for the problem. Here, Tf denotes the time needed to evaluate the function value f(X) for a given X ⊆ V . We also consider the problem of maximizing a given posimodular function. We show that Ω(2) oracle calls are necessary for solving the problem, and that the problem has time complexity Θ(nTf ) when D = {0, 1, . . . , d} is the range of f for some constant d. Keyword: Posimodular function, Algorithm, Horn CNF, Extreme sets