18,328 research outputs found
Covering problems in edge- and node-weighted graphs
This paper discusses the graph covering problem in which a set of edges in an
edge- and node-weighted graph is chosen to satisfy some covering constraints
while minimizing the sum of the weights. In this problem, because of the large
integrality gap of a natural linear programming (LP) relaxation, LP rounding
algorithms based on the relaxation yield poor performance. Here we propose a
stronger LP relaxation for the graph covering problem. The proposed relaxation
is applied to designing primal-dual algorithms for two fundamental graph
covering problems: the prize-collecting edge dominating set problem and the
multicut problem in trees. Our algorithms are an exact polynomial-time
algorithm for the former problem, and a 2-approximation algorithm for the
latter problem, respectively. These results match the currently known best
results for purely edge-weighted graphs.Comment: To appear in SWAT 201
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
Nearly-Linear Time LP Solvers and Rounding Algorithms for Scheduling Problems
We study nearly-linear time approximation algorithms for non-preemptive scheduling problems in two settings: the unrelated machine setting, and the identical machine with job precedence constraints setting, under the well-studied objectives such as makespan and weighted completion time. For many problems, we develop nearly-linear time approximation algorithms with approximation ratios matching the current best ones achieved in polynomial time.
Our main technique is linear programming relaxation. For the unrelated machine setting, we formulate mixed packing and covering LP relaxations of nearly-linear size, and solve them approximately using the nearly-linear time solver of Young. For the makespan objective, we develop a rounding algorithm with (2+?)-approximation ratio. For the weighted completion time objective, we prove the LP is as strong as the rectangle LP used by Im and Li, leading to a nearly-linear time (1.45 + ?)-approximation for the problem.
For problems in the identical machine with precedence constraints setting, the precedence constraints can not be formulated as packing or covering constraints. To achieve the nearly-linear running time, we define a polytope for the constraints, and leverage the multiplicative weight update (MWU) method with an oracle which always returns solutions in the polytope
Approximation Algorithms for Covering/Packing Integer Programs
Given matrices A and B and vectors a, b, c and d, all with non-negative
entries, we consider the problem of computing min {c.x: x in Z^n_+, Ax > a, Bx
< b, x < d}. We give a bicriteria-approximation algorithm that, given epsilon
in (0, 1], finds a solution of cost O(ln(m)/epsilon^2) times optimal, meeting
the covering constraints (Ax > a) and multiplicity constraints (x < d), and
satisfying Bx < (1 + epsilon)b + beta, where beta is the vector of row sums
beta_i = sum_j B_ij. Here m denotes the number of rows of A.
This gives an O(ln m)-approximation algorithm for CIP -- minimum-cost
covering integer programs with multiplicity constraints, i.e., the special case
when there are no packing constraints Bx < b. The previous best approximation
ratio has been O(ln(max_j sum_i A_ij)) since 1982. CIP contains the set cover
problem as a special case, so O(ln m)-approximation is the best possible unless
P=NP.Comment: Preliminary version appeared in IEEE Symposium on Foundations of
Computer Science (2001). To appear in Journal of Computer and System Science
Thresholded Covering Algorithms for Robust and Max-Min Optimization
The general problem of robust optimization is this: one of several possible
scenarios will appear tomorrow, but things are more expensive tomorrow than
they are today. What should you anticipatorily buy today, so that the
worst-case cost (summed over both days) is minimized? Feige et al. and
Khandekar et al. considered the k-robust model where the possible outcomes
tomorrow are given by all demand-subsets of size k, and gave algorithms for the
set cover problem, and the Steiner tree and facility location problems in this
model, respectively.
In this paper, we give the following simple and intuitive template for
k-robust problems: "having built some anticipatory solution, if there exists a
single demand whose augmentation cost is larger than some threshold, augment
the anticipatory solution to cover this demand as well, and repeat". In this
paper we show that this template gives us improved approximation algorithms for
k-robust Steiner tree and set cover, and the first approximation algorithms for
k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios
(except for multicut) are almost best possible.
As a by-product of our techniques, we also get algorithms for max-min
problems of the form: "given a covering problem instance, which k of the
elements are costliest to cover?".Comment: 24 page
Partial resampling to approximate covering integer programs
We consider column-sparse covering integer programs, a generalization of set
cover, which have a long line of research of (randomized) approximation
algorithms. We develop a new rounding scheme based on the Partial Resampling
variant of the Lov\'{a}sz Local Lemma developed by Harris & Srinivasan (2019).
This achieves an approximation ratio of , where is the minimum covering
constraint and is the maximum -norm of any column of the
covering matrix (whose entries are scaled to lie in ). When there are
additional constraints on the variable sizes, we show an approximation ratio of
(where is the maximum number
of non-zero entries in any column of the covering matrix). These results
improve asymptotically, in several different ways, over results of Srinivasan
(2006) and Kolliopoulos & Young (2005).
We show nearly-matching inapproximability and integrality-gap lower bounds.
We also show that the rounding process leads to negative correlation among the
variables, which allows us to handle multi-criteria programs
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