242 research outputs found
Probability and Problems in Euclidean Combinatorial Optimization
This article summarizes the current status of several streams of research that deal with the probability theory of problems of combinatorial optimization. There is a particular emphasis on functionals of finite point sets. The most famous example of such functionals is the length associated with the Euclidean traveling salesman problem (TSP), but closely related problems include the minimal spanning tree problem, minimal matching problems and others. Progress is also surveyed on (1) the approximation and determination of constants whose existence is known by subadditive methods, (2) the central limit problems for several functionals closely related to Euclidean functionals, and (3) analogies in the asymptotic behavior between worst-case and expected-case behavior of Euclidean problems. No attempt has been made in this survey to cover the many important applications of probability to linear programming, arrangement searching or other problems that focus on lines or planes
The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study
Consider a network linking the points of a rate- Poisson point process on
the plane. Write \Psi^{\mbox{ave}}(s) for the minimum possible mean length
per unit area of such a network, subject to the constraint that the
route-length between every pair of points is at most times the Euclidean
distance. We give upper and lower bounds on the function
\Psi^{\mbox{ave}}(s), and on the analogous "worst-case" function
\Psi^{\mbox{worst}}(s) where the point configuration is arbitrary subject to
average density one per unit area. Our bounds are numerically crude, but raise
the question of whether there is an exponent such that each function
has as .Comment: 33 page
Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars
A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of Gleaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the “last mile problem.” Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals
Complexity Theory, Game Theory, and Economics: The Barbados Lectures
This document collects the lecture notes from my mini-course "Complexity
Theory, Game Theory, and Economics," taught at the Bellairs Research Institute
of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th
McGill Invitational Workshop on Computational Complexity.
The goal of this mini-course is twofold: (i) to explain how complexity theory
has helped illuminate several barriers in economics and game theory; and (ii)
to illustrate how game-theoretic questions have led to new and interesting
complexity theory, including recent several breakthroughs. It consists of two
five-lecture sequences: the Solar Lectures, focusing on the communication and
computational complexity of computing equilibria; and the Lunar Lectures,
focusing on applications of complexity theory in game theory and economics. No
background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some
recent citations to v1 Revised v3 corrects a few typos in v
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