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

The traveling salesman problem for lines, balls and planes

We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in $\mathbb{R}^d$, for $d\geq 3$) or improvements over previous approximations achievable in comparable times (for unit disks in the plane). \smallskip (I) Given a set of $n$ hyperplanes in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in $O(n)$ time, when $d$ is constant. \smallskip (II) Given a set of $n$ lines in $\mathbb{R}^d$, a TSP tour whose length is at most $O(\log^3 n)$ times the optimal can be computed in polynomial time for all $d$. \smallskip (III) Given a set of $n$ unit balls in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in polynomial time, when $d$ is constant.Comment: 30 pages, 9 figures; final version to appear in ACM Transactions on Algorithm

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-\epsilon)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-\epsilon)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$

A Gap-{ETH}-Tight Approximation Scheme for Euclidean {TSP}

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ of the algorithm by Rao and Smith (STOC 1998). We also show that a $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree

Approximation Algorithms for Geometric Clustering and Touring Problems

Clustering and touring are two fundamental topics in optimization that have been studied extensively and have ``launched a thousand ships''. In this thesis, we study variants of these problems for Euclidean instances, in which clusters often correspond to sensors that are required to cover, measure or localize targets and tours need to visit locations for the purpose of item delivery or data collection. In the first part of the thesis, we focus on the task of sensor placement for environments in which localization is a necessity and in which its quality depends on the relative angle between the target and the pair of sensors observing it. We formulate a new coverage constraint that bounds this angle and consider the problem of placing a small number of sensors that satisfy it in addition to classical ones such as proximity and line-of-sight visibility. We present a general framework that chooses a small number of sensors and approximates the coverage constraint to arbitrary precision. In the second part of the thesis, we consider the task of collecting data from a set of sensors by getting close to them. This corresponds to a well-known generalization of the Traveling Salesman Problem (TSP) called TSP with Neighborhoods, in which we want to compute a shortest tour that visits at least one point from each unit disk centered at a sensor. One approach is based on an observation that relates the optimal solution with the optimal TSP on the sensors. We show that the associated bound can be improved unless we are in certain exceptional circumstances for which we can get better algorithms. Finally, we discuss Maximum Scatter TSP, which asks for a tour that maximizes the length of the shortest edge. While the Euclidean version admits an efficient approximation scheme and the problem is known to be NP-hard in three dimensions or higher, the question of getting a polynomial time algorithm for two dimensions remains open. To this end, we develop a general technique for the case of points concentrated around the boundary of a circle that we believe can be extended to more general cases

Doctor of Philosophy

dissertationThe contributions of this dissertation are centered around designing new algorithms in the general area of sublinear algorithms such as streaming, core sets and sublinear verification, with a special interest in problems arising from data analysis including data summarization, clustering, matrix problems and massive graphs. In the first part, we focus on summaries and coresets, which are among the main techniques for designing sublinear algorithms for massive data sets. We initiate the study of coresets for uncertain data and study coresets for various types of range counting queries on uncertain data. We focus mainly on the indecisive model of locational uncertainty since it comes up frequently in real-world applications when multiple readings of the same object are made. In this model, each uncertain point has a probability density describing its location, defined as $k$ distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncertainty, so that range counting queries can be answered by examining only this subset. For each type of query we provide coreset constructions with approximation-size trade-offs. We show that random sampling can be used to construct each type of coreset, and we also provide significantly improved bounds using discrepancy-based techniques on axis-aligned range queries. In the second part, we focus on designing sublinear-space algorithms for approximate computations on massive graphs. In particular, we consider graph MAXCUT and correlation clustering problems and develop sampling based approaches to construct truly sublinear ($o(n)$) sized coresets for graphs that have polynomial (i.e., $n^{\delta}$ for any $\delta >0$) average degree. Our technique is based on analyzing properties of random induced subprograms of the linear program formulations of the problems. We demonstrate this technique with two examples. Firstly, we present a sublinear sized core set to approximate the value of the MAX CUT in a graph to a $(1+\epsilon)$ factor. To the best of our knowledge, all the known methods in this regime rely crucially on near-regularity assumptions. Secondly, we apply the same framework to construct a sublinear-sized coreset for correlation clustering. Our coreset construction also suggests 2-pass streaming algorithms for computing the MAX CUT and correlation clustering objective values which are left as future work at the time of writing this dissertation. Finally, we focus on streaming verification algorithms as another model for designing sublinear algorithms. We give the first polylog space and sublinear (in number of edges) communication protocols for any streaming verification problems in graphs. We present efficient streaming interactive proofs that can verify maximum matching exactly. Our results cover all flavors of matchings (bipartite/ nonbipartite and weighted). In addition, we also present streaming verifiers for approximate metric TSP and exact triangle counting, as well as for graph primitives such as the number of connected components, bipartiteness, minimum spanning tree and connectivity. In particular, these are the first results for weighted matchings and for metric TSP in any streaming verification model. Our streaming verifiers use only polylogarithmic space while exchanging only polylogarithmic communication with the prover in addition to the output size of the relevant solution. We also initiate a study of streaming interactive proofs (SIPs) for problems in data analysis and present efficient SIPs for some fundamental problems. We present protocols for clustering and shape fitting including minimum enclosing ball (MEB), width of a point set, $k$-centers and $k$-slab problem. We also present protocols for fundamental matrix analysis problems: We provide an improved protocol for rectangular matrix problems, which in turn can be used to verify $k$ (approximate) eigenvectors of an $n \times n$ integer matrix $A$. In general our solutions use polylogarithmic rounds of communication and polylogarithmic total communication and verifier space