We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers a 1 , . . . , a d ) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.
Abstract. Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed "fairly" among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i's share is precisely equal to i's expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), define and survey computational results on other standard solution concepts, and contrast these results with new results regarding the Shapley value. In particular, we give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ± ε) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ± ε)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic.Topic classification: algorithmic game theory, algorithms, computational complexity
Recent research has highlighted the conflict potential of both land deals and climate change mitigation projects, but generally the two phenomena are studied separately and the focus is limited to discrete cases of displacement or contested claims. We argue that research with a broader "landscape" perspective is needed to better understand the complex social, ecological and institutional interactions taking place in sites of land-based climate change projects (such as biofuel production or forest conservation) and large-scale investments (plantations or mines). Research that coproduces knowledge and capacity with local actors, and informs advocacy at multiple policy scales, will contribute better to preventing, resolving or transforming conflicts.
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