15,366 research outputs found
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
Fault Tolerance in Euclidean Committee Selection
In the committee selection problem, the goal is to choose a subset of size k from a set of candidates C that collectively gives the best representation to a set of voters. We consider this problem in Euclidean d-space where each voter/candidate is a point and voters\u27 preferences are implicitly represented by Euclidean distances to candidates. We explore fault-tolerance in committee selection and study the following three variants: (1) given a committee and a set of f failing candidates, find their optimal replacement; (2) compute the worst-case replacement score for a given committee under failure of f candidates; and (3) design a committee with the best replacement score under worst-case failures. The score of a committee is determined using the well-known (min-max) Chamberlin-Courant rule: minimize the maximum distance between any voter and its closest candidate in the committee. Our main results include the following: (1) in one dimension, all three problems can be solved in polynomial time; (2) in dimension d ? 2, all three problems are NP-hard; and (3) all three problems admit a constant-factor approximation in any fixed dimension, and the optimal committee problem has an FPT bicriterion approximation
Fault Tolerance in Euclidean Committee Selection
In the committee selection problem, the goal is to choose a subset of size
from a set of candidates that collectively gives the best
representation to a set of voters. We consider this problem in Euclidean
-space where each voter/candidate is a point and voters' preferences are
implicitly represented by Euclidean distances to candidates. We explore
fault-tolerance in committee selection and study the following three variants:
(1) given a committee and a set of failing candidates, find their optimal
replacement; (2) compute the worst-case replacement score for a given committee
under failure of candidates; and (3) design a committee with the best
replacement score under worst-case failures. The score of a committee is
determined using the well-known (min-max) Chamberlin-Courant rule: minimize the
maximum distance between any voter and its closest candidate in the committee.
Our main results include the following: (1) in one dimension, all three
problems can be solved in polynomial time; (2) in dimension , all
three problems are NP-hard; and (3) all three problems admit a constant-factor
approximation in any fixed dimension, and the optimal committee problem has an
FPT bicriterion approximation.Comment: The paper will appear in the proceedings of ESA 202
Computer Science and Game Theory: A Brief Survey
There has been a remarkable increase in work at the interface of computer
science and game theory in the past decade. In this article I survey some of
the main themes of work in the area, with a focus on the work in computer
science. Given the length constraints, I make no attempt at being
comprehensive, especially since other surveys are also available, and a
comprehensive survey book will appear shortly.Comment: To appear; Palgrave Dictionary of Economic
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