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Moncler Jackets Outlet Uk formulas where logical implication → is
We consider Boolean formulas where logical implication (→) is the only operator and all variables, except at most one (denoted Discount Moncler Jackets Women z), occur at most twice. We show that the problem of determining falsifiability for formulas of this class is NP-complete but if the number of occurrences of z is restricted to be at most k then there is an O(|F|k) algorithm for certifying falsifiability. We show this hierarchy of formulas, indexed on k, is interesting because even lower levels (e.g., k=2) are not subsumed by several well-known polynomial time solvable classes of formulas. Let v be a cooperative (TU) game and v = v1 − v2 be a decomposition of v as a difference of two convex games v1 and v2. Then the core C(v) of the game v has a similar decomposition C(v) = C(v1) ⊖ C(v2), Moncler Jackets Outlet Uk where ⊖ denotes the Minkowski difference. We prove such a decomposition as a consequence of two claims: the core of a game is equal to the superdifferential of its continuation, known as the Choquet integral, and the superdifferential of a difference of two concave functions equals the Minkowski difference of corresponding superdifferentials.