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

Improved Bounds for Metric Capacitated Covering Problems

In the Metric Capacitated Covering (MCC) problem, given a set of balls ? in a metric space P with metric d and a capacity parameter U, the goal is to find a minimum sized subset ?\u27 ? ? and an assignment of the points in P to the balls in ?\u27 such that each point is assigned to a ball that contains it and each ball is assigned with at most U points. MCC achieves an O(log |P|)-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of o(log |P|) even with ? < 3 factor expansion of the balls. Bandyapadhyay et al. [SoCG 2018, DCG 2019] showed that one can obtain an O(1)-approximation for the problem with 6.47 factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound 3 and the upper bound 6.47. In this current work, we show that it is possible to obtain an O(1)-approximation with only 4.24 factor expansion of the balls. We also show a similar upper bound of 5 for a more generalized version of MCC for which the best previously known bound was 9

Approximation algorithms for geometric dispersion

The most basic form of the max-sum dispersion problem (MSD) is as follows: given n points in R^q and an integer k, select a set of k points such that the sum of the pairwise distances within the set is maximal. This is a prominent diversity problem, with wide applications in web search and information retrieval, where one needs to find a small and diverse representative subset of a large dataset. The problem has recently received a great deal of attention in the computational geometry and operations research communities; and since it is NP-hard, research has focused on efficient heuristics and approximation algorithms. Several classes of distance functions have been considered in the literature. Many of the most common distances used in applications are induced by a norm in a real vector space. The focus of this thesis is on MSD over these geometric instances. We provide for it simple and fast polynomial-time approximation schemes (PTASs), as well as improved constant-factor approximation algorithms. We pay special attention to the class of negative-type distances, a class that includes Euclidean and Manhattan distances, among many others. In order to exploit the properties of this class, we apply several techniques and results from the theory of isometric embeddings. We explore the following variations of the MSD problem: matroid and matroid-intersection constraints, knapsack constraints, and the mixed-objective problem that maximizes a combination of the sum of pairwise distances with a submodular monotone function. In addition to approximation algorithms, we present a core-set for geometric instances of low dimension, and we discuss the efficient implementation of some of our algorithms for massive datasets, using the streaming and distributed models of computation

Quadratic maximization under combinatorial constraints and related applications

Motivated primarily by restricted variants of Principal Component Analysis (PCA), we study quadratic maximization problems subject to sparsity, nonnegativity and other combinatorial constraints. Intuitively, a key technical challenge is determining the support of the optimal solution. We develop a method that can surprisingly solve the maximization exactly when the argument matrix of the quadratic objective is positive semidefinite and has constant rank. Our approach relies on a hyper-spherical transformation of the low-rank space and has complexity that scales exponentially in the rank of the input, but polynomially in the ambient dimension. Extending these observations, we describe a simpler {approximation} algorithm based on exploring the low-rank space with an [epsilon]-net, drastically improving the dependence on the ambient dimension, implying a Polynomial Time Approximation Scheme (PTAS) for inputs with rank scaling up to logarithmically in the dimension or sufficiently sharp spectral decay. We discuss extensions of our approach to jointly computing multiple principal components under combinatorial constraints, such as the problem of extracting multiple orthogonal nonnegative components, or sparse components with common or disjoint supports, and related approximate matrix factorization problems. We further extend our quadratic maximization framework to bilinear optimization problems and employ it in the context of specific applications, e.g., to develop a provable approximation algorithm for the NP-hard problem of Bipartite Correlation Clustering (BCC). Real datasets will typically produce covariance matrices that have full rank, rendering our algorithms not applicable. Our approach is to first obtain a low-rank approximation of the input data and subsequently solve the low-rank problem using our framework. Although this approach is not always suitable, from an optimization perspective it yields provable, data-dependent performance bounds that rely on the spectral decay of the input and the employed approximation technique. Interestingly, most real matrices can be well approximated by low-rank surrogates since the eigenvalues display a significant decay. Empirical evaluation shows that our algorithms have excellent performance and in many cases outperform the previous state of the art. Finally, utilizing our framework, we develop algorithms with interesting theoretical guarantees in the context of specific applications, such as approximate Orthogonal Nonnegative Matrix Factorization or Bipartite Correlation Clustering.Electrical and Computer Engineerin

Fast approximation schemes for multi-criteria combinatorial optimization

Cover title.Includes bibliographical references (p. 38-44).by Hershel M. Safer, James B. Orlin

A PTAS for Euclidean TSP with Hyperplane Neighborhoods

In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given a collection of geometric regions in some space. The goal is to output a tour of minimum length that visits at least one point in each region. Even in the Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying more tractable special cases of the problem. In this paper, we focus on the fundamental special case of regions that are hyperplanes in the $d$-dimensional Euclidean space. This case contrasts the much-better understood case of so-called fat regions. While for $d=2$ an exact algorithm with running time $O(n^5)$ is known, settling the exact approximability of the problem for $d=3$ has been repeatedly posed as an open question. To date, only an approximation algorithm with guarantee exponential in $d$ is known, and NP-hardness remains open. For arbitrary fixed $d$, we develop a Polynomial Time Approximation Scheme (PTAS) that works for both the tour and path version of the problem. Our algorithm is based on approximating the convex hull of the optimal tour by a convex polytope of bounded complexity. Such polytopes are represented as solutions of a sophisticated LP formulation, which we combine with the enumeration of crucial properties of the tour. As the approximation guarantee approaches $1$, our scheme adjusts the complexity of the considered polytopes accordingly. In the analysis of our approximation scheme, we show that our search space includes a sufficiently good approximation of the optimum. To do so, we develop a novel and general sparsification technique to transform an arbitrary convex polytope into one with a constant number of vertices and, in turn, into one of bounded complexity in the above sense. Hereby, we maintain important properties of the polytope

Movement-Efficient Sensor Deployment in Wireless Sensor Networks With Limited Communication Range.

We study a mobile wireless sensor network (MWSN) consisting of multiple mobile sensors or robots. Three key factors in MWSNs, sensing quality, energy consumption, and connectivity, have attracted plenty of attention, but the interaction of these factors is not well studied. To take all the three factors into consideration, we model the sensor deployment problem as a constrained source coding problem. %, which can be applied to different coverage tasks, such as area coverage, target coverage, and barrier coverage. Our goal is to find an optimal sensor deployment (or relocation) to optimize the sensing quality with a limited communication range and a specific network lifetime constraint. We derive necessary conditions for the optimal sensor deployment in both homogeneous and heterogeneous MWSNs. According to our derivation, some sensors are idle in the optimal deployment of heterogeneous MWSNs. Using these necessary conditions, we design both centralized and distributed algorithms to provide a flexible and explicit trade-off between sensing uncertainty and network lifetime. The proposed algorithms are successfully extended to more applications, such as area coverage and target coverage, via properly selected density functions. Simulation results show that our algorithms outperform the existing relocation algorithms