140 research outputs found
A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs
We give an approximation algorithm for packing and covering linear programs
(linear programs with non-negative coefficients). Given a constraint matrix
with n non-zeros, r rows, and c columns, the algorithm computes feasible primal
and dual solutions whose costs are within a factor of 1+eps of the optimal cost
in time O((r+c)log(n)/eps^2 + n).Comment: corrected version of FOCS 2007 paper: 10.1109/FOCS.2007.62. Accepted
to Algorithmica, 201
Nearly Linear-Work Algorithms for Mixed Packing/Covering and Facility-Location Linear Programs
We describe the first nearly linear-time approximation algorithms for
explicitly given mixed packing/covering linear programs, and for (non-metric)
fractional facility location. We also describe the first parallel algorithms
requiring only near-linear total work and finishing in polylog time. The
algorithms compute -approximate solutions in time (and work)
, where is the number of non-zeros in the constraint
matrix. For facility location, is the number of eligible client/facility
pairs
Towards More Practical Linear Programming-based Techniques for Algorithmic Mechanism Design
R. Lavy and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for
obtaining truthful-in-expectation mechanisms from linear programming based
approximation algorithms. Due to the use of the Ellipsoid method, a direct
implementation of the method is unlikely to be efficient in practice. We
propose to use the much simpler and usually faster multiplicative weights
update method instead. The simplification comes at the cost of slightly weaker
approximation and truthfulness guarantees
Online Mixed Packing and Covering
In many problems, the inputs arrive over time, and must be dealt with
irrevocably when they arrive. Such problems are online problems. A common
method of solving online problems is to first solve the corresponding linear
program, and then round the fractional solution online to obtain an integral
solution.
We give algorithms for solving linear programs with mixed packing and
covering constraints online. We first consider mixed packing and covering
linear programs, where packing constraints are given offline and covering
constraints are received online. The objective is to minimize the maximum
multiplicative factor by which any packing constraint is violated, while
satisfying the covering constraints. No prior sublinear competitive algorithms
are known for this problem. We give the first such --- a
polylogarithmic-competitive algorithm for solving mixed packing and covering
linear programs online. We also show a nearly tight lower bound.
Our techniques for the upper bound use an exponential penalty function in
conjunction with multiplicative updates. While exponential penalty functions
are used previously to solve linear programs offline approximately, offline
algorithms know the constraints beforehand and can optimize greedily. In
contrast, when constraints arrive online, updates need to be more complex.
We apply our techniques to solve two online fixed-charge problems with
congestion. These problems are motivated by applications in machine scheduling
and facility location. The linear program for these problems is more
complicated than mixed packing and covering, and presents unique challenges. We
show that our techniques combined with a randomized rounding procedure give
polylogarithmic-competitive integral solutions. These problems generalize
online set-cover, for which there is a polylogarithmic lower bound. Hence, our
results are close to tight
Towards More Practical Linear Programming-based Techniques for Algorithmic Mechanism Design
R. Lavy and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-in-expectation mechanisms from linear programming based approximation algorithms. Due to the use of the Ellipsoid method, a direct implementation of the method is unlikely to be efficient in practice. We propose to use the much simpler and usually faster multiplicative weights update method instead. The simplification comes at the cost of slightly weaker approximation and truthfulness guarantees
On Randomized Fictitious Play for Approximating Saddle Points Over Convex Sets
Given two bounded convex sets X\subseteq\RR^m and Y\subseteq\RR^n, specified by membership oracles, and a continuous convex-concave function F:X\times Y\to\RR, we consider the problem of computing an \eps-approximate saddle point, that is, a pair such that \sup_{y\in Y} F(x^*,y)\le \inf_{x\in X}F(x,y^*)+\eps. Grigoriadis and Khachiyan (1995) gave a simple randomized variant of fictitious play for computing an \eps-approximate saddle point for matrix games, that is, when is bilinear and the sets and are simplices. In this paper, we extend their method to the general case. In particular, we show that, for functions of constant "width", an \eps-approximate saddle point can be computed using O^*(\frac{(n+m)}{\eps^2}\ln R) random samples from log-concave distributions over the convex sets and . It is assumed that and have inscribed balls of radius and circumscribing balls of radius . As a consequence, we obtain a simple randomized polynomial-time algorithm that computes such an approximation faster than known methods for problems with bounded width and when \eps \in (0,1) is a fixed, but arbitrarily small constant. Our main tool for achieving this result is the combination of the randomized fictitious play with the recently developed results on sampling from convex sets
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