An ETH-Tight Exact Algorithm for Euclidean TSP

We study exact algorithms for {\sc Euclidean TSP} in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was presented for any $d\geq 2$. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on {\sc Euclidean TSP}, except for a lower bound stating that the problem admits no $2^{O(n^{1-1/d-\epsilon})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of {\sc Euclidean TSP} by giving a $2^{O(n^{1-1/d})}$ algorithm and by showing that a $2^{o(n^{1-1/d})}$ algorithm does not exist unless ETH fails.Comment: To appear in FOCS 201

Euclidean TSP with few inner points in linear space

Given a set of $n$ points in the Euclidean plane, such that just $k$ points are strictly inside the convex hull of the whole set, we want to find the shortest tour visiting every point. The fastest known algorithm for the version when $k$ is significantly smaller than $n$, i.e., when there are just few inner points, works in $O(k^{11\sqrt{k}} k^{1.5} n^{3})$ time [Knauer and Spillner, WG 2006], but also requires space of order $k^{c\sqrt{k}}n^{2}$. The best linear space algorithm takes $O(k! k n)$ time [Deineko, Hoffmann, Okamoto, Woeginer, Oper. Res. Lett. 34(1), 106-110]. We construct a linear space $O(nk^2+k^{O(\sqrt{k})})$ time algorithm. The new insight is extending the known divide-and-conquer method based on planar separators with a matching-based argument to shrink the instance in every recursive call. This argument also shows that the problem admits a quadratic bikernel.Comment: under submissio

Approximation Complexity of Optimization Problems : Structural Foundations and Steiner Tree Problems

In this thesis we study the approximation complexity of the Steiner Tree Problem and related problems as well as foundations in structural complexity theory. The Steiner Tree Problem is one of the most fundamental problems in combinatorial optimization. It asks for a shortest connection of a given set of points in an edge-weighted graph. This problem and its numerous variants have applications ranging from electrical engineering, VLSI design and transportation networks to internet routing. It is closely connected to the famous Traveling Salesman Problem and serves as a benchmark problem for approximation algorithms. We give a survey on the Steiner tree Problem, obtaining lower bounds for approximability of the (1,2)-Steiner Tree Problem by combining hardness results of Berman and Karpinski with reduction methods of Bern and Plassmann. We present approximation algorithms for the Steiner Forest Problem in graphs and bounded hypergraphs, the Prize Collecting Steiner Tree Problem and related problems where prizes are given for pairs of terminals. These results are based on the Primal-Dual method and the Local Ratio framework of Bar-Yehuda. We study the Steiner Network Problem and obtain combinatorial approximation algorithms with reasonable running time for two special cases, namely the Uniform Uncapacitated Case and the Prize Collecting Uniform Uncapacitated Case. For the general case, Jain's algorithms obtains an approximation ratio of 2, based on the Ellipsoid Method. We obtain polynomial time approximation schemes for the Dense Prize Collecting Steiner Tree Problem, Dense k-Steiner Problem and the Dense Class Steiner Tree Problem based on the methods of Karpinski and Zelikovsky for approximating the Dense Steiner Tree Problem. Motivated by the question which parameters make the Steiner Tree problem hard to solve, we make an excurs into Fixed Parameter Complexity, focussing on structural aspects of the W-Hierarchy. We prove a Speedup Theorem for the classes FPT and SP and versions if Levin's Lower Bound Theorem for the class SP as well as for Randomized Space Complexity. Starting from the approximation schemes for the dense Steiner Tree problems, we deal with the efficiency of polynomial time approximation schemes in general. We separate the class EPTAS from PTAS under some reasonable complexity theoretic assumption. The same separation was achieved by Cesaty and Trevisan under some assumtion from Fixed Parameter Complexity. We construct an oracle under which our assumtion holds but that of Cesati and Trevisan does not, which implies that using relativizing proof techniques one cannot show that our assumption implies theirs

The Bane of Low-Dimensionality Clustering

In this paper, we give a conditional lower bound of $n^{\Omega(k)}$ on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time $n^{O(k^{1-1/d})}$ or $2^{n^{1-1/d}}$ in d dimensions, our work shows that widely-used clustering objectives have a lower bound of $n^{\Omega(k)}$, even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than $n^{o(\sqrt{k})}$, and provide a matching upper bound of $n^{O(\sqrt{k})}$. The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity

"Interpolated Factored Green Function" Method for accelerated solution of Scattering Problems

This paper presents a novel {\em Interpolated Factored Green Function} method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of $\mathcal{O}(N\log N)$ operations for an $N$-point surface mesh. The IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green function {\em analytic factor}, which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. Only a serial implementation of the algorithm is considered in this paper, however, whose efficiency in terms of memory and speed is illustrated by means of a variety of numerical experiments -- including a 43 min., single-core operator evaluation (on 10 GB of peak memory), with a relative error of $1.5\times 10^{-2}$, for a problem of acoustic size of 512$\lambda$.Comment: 38 pages, 16 figures, 7 tables, 1 algorith

“Interpolated Factored Green Function” Method for accelerated solution of Scattering Problems

This paper presents a novel Interpolated Factored Green Function method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O(N log N) operations for an N-point surface mesh. The IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green function analytic factor, which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. Only a serial implementation of the algorithm is considered in this paper, however, whose efficiency in terms of memory and speed is illustrated by means of a variety of numerical experiments—including a 43 min., single-core operator evaluation (on 10 GB of peak memory), with a relative error of 1.5×10⁻², for a problem of acoustic size of 512λ. wavelengths

Algorithm engineering in geometric network planning and data mining

The geometric nature of computational problems provides a rich source of solution strategies as well as complicating obstacles. This thesis considers three problems in the context of geometric network planning, data mining and spherical geometry. Geometric Network Planning: In the d-dimensional Generalized Minimum Manhattan Network problem (d-GMMN) one is interested in finding a minimum cost rectilinear network N connecting a given set of n pairs of points in ℝ^d such that each pair is connected in N via a shortest Manhattan path. The decision version of this optimization problem is known to be NP-hard. The best known upper bound is an O(log^{d+1} n) approximation for d>2 and an O(log n) approximation for 2-GMMN. In this work we provide some more insight in, whether the problem admits constant factor approximations in polynomial time. We develop two new algorithms, a `scale-diversity aware' algorithm with an O(D) approximation guarantee for 2-GMMN. Here D is a measure for the different `scales' that appear in the input, D ∈ O(log n) but potentially much smaller, depending on the problem instance. The other algorithm is based on a primal-dual scheme solving a more general, combinatorial problem - which we call Path Cover. On 2-GMMN it performs well in practice with good a posteriori, instance-based approximation guarantees. Furthermore, it can be extended to deal with obstacle avoiding requirements. We show that the Path Cover problem is at least as hard to approximate as the Hitting Set problem. Moreover, we show that solutions of the primal-dual algorithm are 4ω^2 approximations, where ω ≤ n denotes the maximum overlap of a problem instance. This implies that a potential proof of O(1)-inapproximability for 2-GMMN requires gadgets of many different scales and non-constant overlap in the construction. Geometric Map Matching for Heterogeneous Data: For a given sequence of location measurements, the goal of the geometric map matching is to compute a sequence of movements along edges of a spatially embedded graph which provides a `good explanation' for the measurements. The problem gets challenging as real world data, like traces or graphs from the OpenStreetMap project, does not exhibit homogeneous data quality. Graph details and errors vary in areas and each trace has changing noise and precision. Hence, formalizing what a `good explanation' is becomes quite difficult. We propose a novel map matching approach, which locally adapts to the data quality by constructing what we call dominance decompositions. While our approach is computationally more expensive than previous approaches, our experiments show that it allows for high quality map matching, even in presence of highly variable data quality without parameter tuning. Rational Points on the Unit Spheres: Each non-zero point in ℝ^d identifies a closest point x on the unit sphere S^{d-1}. We are interested in computing an ε-approximation y ∈ ℚ^d for x, that is exactly on S^{d-1} and has low bit-size. We revise lower bounds on rational approximations and provide explicit spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in ℝ^2 and ℝ^3. However, we show how to construct a rational point with denominators of at most 10(d-1)/ε^2 for any given ε ∈ (0, 1/8], improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation. Our open-source implementation and experiments demonstrate the practicality of our approach in the context of massive data sets, geo-referenced by latitude and longitude values.Die geometrische Gestalt von Berechnungsproblemen liefert vielfältige Lösungsstrategieen aber auch Hindernisse. Diese Arbeit betrachtet drei Probleme im Gebiet der geometrischen Netzwerk Planung, des geometrischen Data Minings und der sphärischen Geometrie. Geometrische Netzwerk Planung: Im d-dimensionalen Generalized Minimum Manhattan Network Problem (d-GMMN) möchte man ein günstigstes geradliniges Netzwerk finden, welches jedes der gegebenen n Punktepaare aus ℝ^d mit einem kürzesten Manhattan Pfad verbindet. Es ist bekannt, dass die Entscheidungsvariante dieses Optimierungsproblems NP-hart ist. Die beste bekannte obere Schranke ist eine O(log^{d+1} n) Approximation für d>2 und eine O(log n) Approximation für 2-GMMN. Durch diese Arbeit geben wir etwas mehr Einblick, ob das Problem eine Approximation mit konstantem Faktor in polynomieller Zeit zulässt. Wir entwickeln zwei neue Algorithmen. Ersterer nutzt die `Skalendiversität' und hat eine O(D) Approximationsgüte für 2-GMMN. Hierbei ist D ein Maß für die in Eingaben auftretende `Skalen'. D ∈ O(log n), aber potentiell deutlichen kleiner für manche Problem Instanzen. Der andere Algorithmus basiert auf einem Primal-Dual Schema zur Lösung eines allgemeineren, kombinatorischen Problems, welches wir Path Cover nennen. Die praktisch erzielten a posteriori Approximationsgüten auf Instanzen von 2-GMMN verhalten sich gut. Dieser Algorithmus kann für Netzwerk Planungsprobleme mit Hindernis-Anforderungen angepasst werden. Wir zeigen, dass das Path Cover Problem mindestens so schwierig zu approximieren ist wie das Hitting Set Problem. Darüber hinaus zeigen wir, dass Lösungen des Primal-Dual Algorithmus 4ω^2 Approximationen sind, wobei ω ≤ n die maximale Überlappung einer Probleminstanz bezeichnet. Daher müssen potentielle Beweise, die konstante Approximationen für 2-GMMN ausschließen möchten, Instanzen mit vielen unterschiedlichen Skalen und nicht konstanter Überlappung konstruieren. Geometrisches Map Matching für heterogene Daten: Für eine gegebene Sequenz von Positionsmessungen ist das Ziel des geometrischen Map Matchings eine Sequenz von Bewegungen entlang Kanten eines räumlich eingebetteten Graphen zu finden, welche eine `gute Erklärung' für die Messungen ist. Das Problem wird anspruchsvoll da reale Messungen, wie beispielsweise Traces oder Graphen des OpenStreetMap Projekts, keine homogene Datenqualität aufweisen. Graphdetails und -fehler variieren in Gebieten und jeder Trace hat wechselndes Rauschen und Messgenauigkeiten. Zu formalisieren, was eine `gute Erklärung' ist, wird dadurch schwer. Wir stellen einen neuen Map Matching Ansatz vor, welcher sich lokal der Datenqualität anpasst indem er sogenannte Dominance Decompositions berechnet. Obwohl unser Ansatz teurer im Rechenaufwand ist, zeigen unsere Experimente, dass qualitativ hochwertige Map Matching Ergebnisse auf hoch variabler Datenqualität erzielbar sind ohne vorher Parameter kalibrieren zu müssen. Rationale Punkte auf Einheitssphären: Jeder, von Null verschiedene, Punkt in ℝ^d identifiziert einen nächsten Punkt x auf der Einheitssphäre S^{d-1}. Wir suchen eine ε-Approximation y ∈ ℚ^d für x zu berechnen, welche exakt auf S^{d-1} ist und niedrige Bit-Größe hat. Wir wiederholen untere Schranken an rationale Approximationen und liefern explizite, sphärische Instanzen. Wir beweisen, dass Floating-Point Zahlen nur triviale Lösungen zur Sphären-Gleichung in ℝ^2 und ℝ^3 liefern können. Jedoch zeigen wir die Konstruktion eines rationalen Punktes mit Nennern die maximal 10(d-1)/ε^2 sind für gegebene ε ∈ (0, 1/8], was ein bekanntes Resultat verbessert. Darüber hinaus profitiert die Methode von Algorithmen für simultane Diophantische Approximationen. Unsere quell-offene Implementierung und die Experimente demonstrieren die Praktikabilität unseres Ansatzes für sehr große, durch geometrische Längen- und Breitengrade referenzierte, Datensätze