250 research outputs found
Capacitated Covering Problems in Geometric Spaces
In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B\u27 subseteq B of balls and assign each point in P to some ball in B\u27 that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B\u27.
We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only (1+epsilon) factor expansion is sufficient for any epsilon > 0, with the approximation factor being a polynomial in 1/epsilon. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
Coresets for Clustering with General Assignment Constraints
Designing small-sized \emph{coresets}, which approximately preserve the costs
of the solutions for large datasets, has been an important research direction
for the past decade. We consider coreset construction for a variety of general
constrained clustering problems. We significantly extend and generalize the
results of a very recent paper (Braverman et al., FOCS'22), by demonstrating
that the idea of hierarchical uniform sampling (Chen, SICOMP'09; Braverman et
al., FOCS'22) can be applied to efficiently construct coresets for a very
general class of constrained clustering problems with general assignment
constraints, including capacity constraints on cluster centers, and assignment
structure constraints for data points (modeled by a convex body .
Our main theorem shows that a small-sized -coreset exists as long
as a complexity measure of the structure
constraint, and the \emph{covering exponent}
for metric space are bounded. The complexity measure
for convex body is the Lipschitz
constant of a certain transportation problem constrained in ,
called \emph{optimal assignment transportation problem}. We prove nontrivial
upper bounds of for various polytopes, including
the general matroid basis polytopes, and laminar matroid polytopes (with better
bound). As an application of our general theorem, we construct the first
coreset for the fault-tolerant clustering problem (with or without capacity
upper/lower bound) for the above metric spaces, in which the fault-tolerance
requirement is captured by a uniform matroid basis polytope
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii
Clustering with capacity constraints is a fundamental problem that attracted
significant attention throughout the years. In this paper, we give the first
FPT constant-factor approximation algorithm for the problem of clustering
points in a general metric into clusters to minimize the sum of cluster
radii, subject to non-uniform hard capacity constraints. In particular, we give
a -approximation algorithm that runs in time. When capacities are uniform, we obtain the following improved
approximation bounds: A (4 + )-approximation with running time
, which significantly improves over the FPT
28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 +
)-approximation with running time and a -approximation with running
time in the Euclidean space; and a (1 +
)-approximation in the Euclidean space with running time
if we are allowed to violate
the capacities by (1 + )-factor. We complement this result by showing
that there is no (1 + )-approximation algorithm running in time
, if any capacity violation is not allowed.Comment: Full version of a paper accepted to SoCG 202
Non-uniform geometric set cover and scheduling on multiple machines
We consider the following general scheduling problem studied recently by Moseley [27]. There are n jobs, all released at time 0, where job j has size pj and an associated arbitrary non-decreasing cost function fj of its completion time.
A Technique for Obtaining True Approximations for -Center with Covering Constraints
There has been a recent surge of interest in incorporating fairness aspects
into classical clustering problems. Two recently introduced variants of the
-Center problem in this spirit are Colorful -Center, introduced by
Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the
Fair Robust -Center problem introduced by Harris, Pensyl, Srinivasan, and
Trinh. To address fairness aspects, these models, compared to traditional
-Center, include additional covering constraints. Prior approximation
results for these models require to relax some of the normally hard
constraints, like the number of centers to be opened or the involved covering
constraints, and therefore, only obtain constant-factor pseudo-approximations.
In this paper, we introduce a new approach to deal with such covering
constraints that leads to (true) approximations, including a -approximation
for Colorful -Center with constantly many colors---settling an open question
raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan---and a
-approximation for Fair Robust -Center, for which the existence of a
(true) constant-factor approximation was also open. We complement our results
by showing that if one allows an unbounded number of colors, then Colorful
-Center admits no approximation algorithm with finite approximation
guarantee, assuming that . Moreover, under the
Exponential Time Hypothesis, the problem is inapproximable if the number of
colors grows faster than logarithmic in the size of the ground set
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Covering Problems via Structural Approaches
The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible.
In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide:
- An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC.
- A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure.
- A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity.
- Applications of the above to various capacitated covering problems via linear programming strengthening and rounding.
In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability
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