296 research outputs found
Matroid and Knapsack Center Problems
In the classic -center problem, we are given a metric graph, and the
objective is to open nodes as centers such that the maximum distance from
any vertex to its closest center is minimized. In this paper, we consider two
important generalizations of -center, the matroid center problem and the
knapsack center problem. Both problems are motivated by recent content
distribution network applications. Our contributions can be summarized as
follows:
1. We consider the matroid center problem in which the centers are required
to form an independent set of a given matroid. We show this problem is NP-hard
even on a line. We present a 3-approximation algorithm for the problem on
general metrics. We also consider the outlier version of the problem where a
given number of vertices can be excluded as the outliers from the solution. We
present a 7-approximation for the outlier version.
2. We consider the (multi-)knapsack center problem in which the centers are
required to satisfy one (or more) knapsack constraint(s). It is known that the
knapsack center problem with a single knapsack constraint admits a
3-approximation. However, when there are at least two knapsack constraints, we
show this problem is not approximable at all. To complement the hardness
result, we present a polynomial time algorithm that gives a 3-approximate
solution such that one knapsack constraint is satisfied and the others may be
violated by at most a factor of . We also obtain a 3-approximation
for the outlier version that may violate the knapsack constraint by
.Comment: A preliminary version of this paper is accepted to IPCO 201
Small Space Stream Summary for Matroid Center
In the matroid center problem, which generalizes the k-center problem, we need to pick a set of centers that is an independent set of a matroid with rank r. We study this problem in streaming, where elements of the ground set arrive in the stream. We first show that any randomized one-pass streaming algorithm that computes a better than Delta-approximation for partition-matroid center must use Omega(r^2) bits of space, where Delta is the aspect ratio of the metric and can be arbitrarily large. This shows a quadratic separation between matroid center and k-center, for which the Doubling algorithm [Charikar et al., 1997] gives an 8-approximation using O(k)-space and one pass. To complement this, we give a one-pass algorithm for matroid center that stores at most O(r^2 log(1/epsilon)/epsilon) points (viz., stream summary) among which a (7+epsilon)-approximate solution exists, which can be found by brute force, or a (17+epsilon)-approximation can be found with an efficient algorithm. If we are allowed a second pass, we can compute a (3+epsilon)-approximation efficiently.
We also consider the problem of matroid center with z outliers and give a one-pass algorithm that outputs a set of O((r^2+rz)log(1/epsilon)/epsilon) points that contains a (15+epsilon)-approximate solution. Our techniques extend to knapsack center and knapsack center with z outliers in a straightforward way, and we get algorithms that use space linear in the size of a largest feasible set (as opposed to quadratic space for matroid center)
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
Consider the following problem: given a set system (U,I) and an edge-weighted
graph G = (U, E) on the same universe U, find the set A in I such that the
Steiner tree cost with terminals A is as large as possible: "which set in I is
the most difficult to connect up?" This is an example of a max-min problem:
find the set A in I such that the value of some minimization (covering) problem
is as large as possible.
In this paper, we show that for certain covering problems which admit good
deterministic online algorithms, we can give good algorithms for max-min
optimization when the set system I is given by a p-system or q-knapsacks or
both. This result is similar to results for constrained maximization of
submodular functions. Although many natural covering problems are not even
approximately submodular, we show that one can use properties of the online
algorithm as a surrogate for submodularity.
Moreover, we give stronger connections between max-min optimization and
two-stage robust optimization, and hence give improved algorithms for robust
versions of various covering problems, for cases where the uncertainty sets are
given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and
http://arxiv.org/abs/0912.1045 appeared in ICALP 201
Dependent randomized rounding for clustering and partition systems with knapsack constraints
Clustering problems are fundamental to unsupervised learning. There is an
increased emphasis on fairness in machine learning and AI; one representative
notion of fairness is that no single demographic group should be
over-represented among the cluster-centers. This, and much more general
clustering problems, can be formulated with "knapsack" and "partition"
constraints. We develop new randomized algorithms targeting such problems, and
study two in particular: multi-knapsack median and multi-knapsack center. Our
rounding algorithms give new approximation and pseudo-approximation algorithms
for these problems. One key technical tool, which may be of independent
interest, is a new tail bound analogous to Feige (2006) for sums of random
variables with unbounded variances. Such bounds are very useful in inferring
properties of large networks using few samples
Constant Approximation for -Median and -Means with Outliers via Iterative Rounding
In this paper, we present a new iterative rounding framework for many
clustering problems. Using this, we obtain an -approximation algorithm for -median with outliers, greatly
improving upon the large implicit constant approximation ratio of Chen [Chen,
SODA 2018]. For -means with outliers, we give an -approximation, which is the first -approximation for
this problem. The iterative algorithm framework is very versatile; we show how
it can be used to give - and -approximation
algorithms for matroid and knapsack median problems respectively, improving
upon the previous best approximations ratios of [Swamy, ACM Trans.
Algorithms] and [Byrka et al, ESA 2015].
The natural LP relaxation for the -median/-means with outliers problem
has an unbounded integrality gap. In spite of this negative result, our
iterative rounding framework shows that we can round an LP solution to an
almost-integral solution of small cost, in which we have at most two
fractionally open facilities. Thus, the LP integrality gap arises due to the
gap between almost-integral and fully-integral solutions. Then, using a
pre-processing procedure, we show how to convert an almost-integral solution to
a fully-integral solution losing only a constant-factor in the approximation
ratio. By further using a sparsification technique, the additive factor loss
incurred by the conversion can be reduced to any
A Lottery Model for Center-Type Problems with Outliers
In this paper, we give tight approximation algorithms for the k-center and matroid center problems with outliers. Unfairness arises naturally in this setting: certain clients could always be considered as outliers. To address this issue, we introduce a lottery model in which each client is allowed to submit a parameter indicating the lower-bound on the probability that it should be covered and we look for a random solution that satisfies all the given requests. Out techniques include a randomized rounding procedure to round a point inside a matroid intersection polytope to a basis plus at most one extra item such that all marginal probabilities are preserved and such that a certain linear function of the variables does not decrease in the process with probability one
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