2,400 research outputs found
Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits
Initially developed for the min-knapsack problem, the knapsack cover
inequalities are used in the current best relaxations for numerous
combinatorial optimization problems of covering type. In spite of their
widespread use, these inequalities yield linear programming (LP) relaxations of
exponential size, over which it is not known how to optimize exactly in
polynomial time. In this paper we address this issue and obtain LP relaxations
of quasi-polynomial size that are at least as strong as that given by the
knapsack cover inequalities.
For the min-knapsack cover problem, our main result can be stated formally as
follows: for any , there is a -size LP relaxation with an integrality gap of at most ,
where is the number of items. Prior to this work, there was no known
relaxation of subexponential size with a constant upper bound on the
integrality gap.
Our construction is inspired by a connection between extended formulations
and monotone circuit complexity via Karchmer-Wigderson games. In particular,
our LP is based on -depth monotone circuits with fan-in~ for
evaluating weighted threshold functions with inputs, as constructed by
Beimel and Weinreb. We believe that a further understanding of this connection
may lead to more positive results complementing the numerous lower bounds
recently proved for extended formulations.Comment: 21 page
Noise Sensitivity of Boolean Functions and Applications to Percolation
It is shown that a large class of events in a product probability space are
highly sensitive to noise, in the sense that with high probability, the
configuration with an arbitrary small percent of random errors gives almost no
prediction whether the event occurs. On the other hand, weighted majority
functions are shown to be noise-stable. Several necessary and sufficient
conditions for noise sensitivity and stability are given.
Consider, for example, bond percolation on an by grid. A
configuration is a function that assigns to every edge the value 0 or 1. Let
be a random configuration, selected according to the uniform measure.
A crossing is a path that joins the left and right sides of the rectangle, and
consists entirely of edges with . By duality, the probability
for having a crossing is 1/2. Fix an . For each edge , let
with probability , and
with probability , independently of the
other edges. Let be the probability for having a crossing in
, conditioned on . Then for all sufficiently large,
.Comment: To appear in Inst. Hautes Etudes Sci. Publ. Mat
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
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