1,308 research outputs found
Bounds for the diameter of the weight polytope
A weighted game or a threshold function in general admits different weighted
representations even if the sum of non-negative weights is fixed to one. Here
we study bounds for the diameter of the corresponding weight polytope. It turns
out that the diameter can be upper bounded in terms of the maximum weight and
the quota or threshold. We apply those results to approximation results between
power distributions, given by power indices, and weights.Comment: 16 pages; typos corrected; arXiv admin note: text overlap with
arXiv:1802.0049
Not all simplicial polytopes are weakly vertex-decomposable
In 1980 Provan and Billera defined the notion of weak -decomposability for
pure simplicial complexes. They showed the diameter of a weakly
-decomposable simplicial complex is bounded above by a polynomial
function of the number of -faces in and its dimension. For weakly
0-decomposable complexes, this bound is linear in the number of vertices and
the dimension. In this paper we exhibit the first examples of non-weakly
0-decomposable simplicial polytopes
Convex Combinatorial Optimization
We introduce the convex combinatorial optimization problem, a far reaching
generalization of the standard linear combinatorial optimization problem. We
show that it is strongly polynomial time solvable over any edge-guaranteed
family, and discuss several applications
Computational Geometry Column 34
Problems presented at the open-problem session of the 14th Annual ACM
Symposium on Computational Geometry are listed
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Parametric shortest-path algorithms via tropical geometry
We study parameterized versions of classical algorithms for computing
shortest-path trees. This is most easily expressed in terms of tropical
geometry. Applications include shortest paths in traffic networks with variable
link travel times.Comment: 24 pages and 8 figure
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