14 research outputs found
Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications
We consider the matroid median problem, wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation by Krishnaswamy et al. We illustrate the power and versatility of our techniques by deriving: (a) an 8-approximation for the two-matroid median problem, a generalization of matroid median that we introduce involving two matroids; and (b) a 24-approximation algorithm for matroid median with penalties, which is a vast improvement over the 360-approximation obtained by Krishnaswamy et al. We show that a variety of seemingly disparate facility-location problems considered in the literature -- data placement problem, mobile facility location, k-median forest, metric uniform minimum-latency UFL -- in fact reduce to the matroid median or two-matroid median problems, and thus obtain improved approximation guarantees for all these problems. Our techniques also yield an improvement for the knapsack median problem
Diversity-aware -median : Clustering with fair center representation
We introduce a novel problem for diversity-aware clustering. We assume that
the potential cluster centers belong to a set of groups defined by protected
attributes, such as ethnicity, gender, etc. We then ask to find a minimum-cost
clustering of the data into clusters so that a specified minimum number of
cluster centers are chosen from each group. We thus require that all groups are
represented in the clustering solution as cluster centers, according to
specified requirements. More precisely, we are given a set of clients , a
set of facilities \pazocal{F}, a collection
of facility groups F_i \subseteq \pazocal{F}, budget , and a set of
lower-bound thresholds , one for each group in
. The \emph{diversity-aware -median problem} asks to find a set
of facilities in \pazocal{F} such that , that
is, at least centers in are from group , and the -median cost
is minimized. We show that in the
general case where the facility groups may overlap, the diversity-aware
-median problem is \np-hard, fixed-parameter intractable, and inapproximable
to any multiplicative factor. On the other hand, when the facility groups are
disjoint, approximation algorithms can be obtained by reduction to the
\emph{matroid median} and \emph{red-blue median} problems. Experimentally, we
evaluate our approximation methods for the tractable cases, and present a
relaxation-based heuristic for the theoretically intractable case, which can
provide high-quality and efficient solutions for real-world datasets.Comment: To appear in ECML-PKDD 202
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
Local search heuristics for the mobile facility location problem
a b s t r a c t In the mobile facility location problem (MFLP), one seeks to relocate (or move) a set of existing facilities and assign clients to these facilities so that the sum of facility movement costs and the client travel costs (each to its assigned facility) is minimized. This paper studies formulations and develops local search heuristics for the MFLP. First, we develop an integer programming (IP) formulation for the MFLP by observing that for a given set of facility destinations the problem may be decomposed into two polynomially solvable subproblems. This IP formulation is quite compact in terms of the number of nonzero coefficients in the constraint matrix and the number of integer variables; and allows for the solution of large-scale MFLP instances. Using the decomposition observation, we propose two local search neighborhoods for the MFLP. We report on extensive computational tests of the new IP formulation and local search heuristics on a large range of instances. These tests demonstrate that the proposed formulation and local search heuristics significantly outperform the existing formulation and a previously developed local search heuristic for the problem