29,530 research outputs found
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
Online Mixed Packing and Covering
Recent work has shown that the classical framework of
solving optimization problems by obtaining a fractional
solution to a linear program (LP) and rounding it to
an integer solution can be extended to the online setting
using primal-dual techniques. The success of this
new framework for online optimization can be gauged
from the fact that it has led to progress in several longstanding open questions. However, to the best of our
knowledge, this framework has previously been applied
to LPs containing only packing or only covering constraints,
or minor variants of these. We extend this
framework in a fundamental way by demonstrating that
it can be used to solve mixed packing and covering LPs
online, 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. Our results represent the
first algorithm that obtains a polylogarithmic competitive
ratio for solving mixed LPs online.
We then consider two canonical examples of mixed
LPs: unrelated machine scheduling with startup costs,
and capacity constrained facility location. We use ideas
generated from our result for mixed packing and covering
to obtain polylogarithmic-competitive algorithms
for these problems. We also give lower bounds to show
that the competitive ratios of our algorithms are nearly
tight
Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering
We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest (EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF, we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance, we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal, we need to augment our solution subgraph to connect the new terminal to the root. The goal is to minimize the total weight of the solution while respecting the degree bounds on the vertices. In the offline setting, edge-weighted degree-bounded Steiner tree (EW-DB-ST) and its many variations have been extensively studied since early eighties. Unfortunately, the recent advancements in the online network design problems are inherently difficult to adapt for degree-bounded problems. In particular, it is not known whether the fractional solution obtained by standard primal-dual techniques for mixed packing/covering LPs can be rounded online. In contrast, in this paper we obtain our result by using structural properties of the optimal solution, and reducing the EW-DB-SF problem to an exponential-size mixed packing/covering integer program in which every variable appears only once in covering constraints. We then design a generic integral algorithm for solving this restricted family of IPs.
As mentioned above, we demonstrate a new technique for solving mixed packing/covering integer programs. Define the covering frequency k of a program as the maximum number of covering constraints in which a variable can participate. Let m denote the number of packing constraints. We design an online deterministic integral algorithm with competitive ratio of O(k*log(m)) for the mixed packing/covering integer programs. We prove the tightness of our result by providing a matching lower bound for any randomized algorithm. We note that our solution solely depends on m and k. Indeed, there can be exponentially many variables. Furthermore, our algorithm directly provides an integral solution, even if the integrality gap of the program is unbounded. We believe this technique can be used as an interesting alternative for the standard primal-dual techniques in solving online problems
How the Experts Algorithm Can Help Solve LPs Online
We consider the problem of solving packing/covering LPs online, when the
columns of the constraint matrix are presented in random order. This problem
has received much attention and the main focus is to figure out how large the
right-hand sides of the LPs have to be (compared to the entries on the
left-hand side of the constraints) to allow -approximations
online. It is known that the right-hand sides have to be times the left-hand sides, where is the number of constraints.
In this paper we give a primal-dual algorithm that achieve this bound for
mixed packing/covering LPs. Our algorithms construct dual solutions using a
regret-minimizing online learning algorithm in a black-box fashion, and use
them to construct primal solutions. The adversarial guarantee that holds for
the constructed duals helps us to take care of most of the correlations that
arise in the algorithm; the remaining correlations are handled via martingale
concentration and maximal inequalities. These ideas lead to conceptually simple
and modular algorithms, which we hope will be useful in other contexts.Comment: An extended abstract appears in the 22nd European Symposium on
Algorithms (ESA 2014
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
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