In pursuit of linear complexity in discrete and computational geometry

Many computational problems arise naturally from geometric data. In this thesis, we consider three such problems: (i) distance optimization problems over point sets, (ii) computing contour trees over simplicial meshes, and (iii) bounding the expected complexity of weighted Voronoi diagrams. While these topics are broad, here the focus is on identifying structure which implies linear (or near linear) algorithmic and descriptive complexity. The first topic we consider is in geometric optimization. More specifically, we define a large class of distance problems, for which we provide linear time exact or approximate solutions. Roughly speaking, the class of problems facilitate either clustering together close points (i.e. netting) or throwing out outliers (i.e pruning), allowing for successively smaller summaries of the relevant information in the input. A surprising number of classical geometric optimization problems are unified under this framework, including finding the optimal k-center clustering, the kth ranked distance, the kth heaviest edge of the MST, the minimum radius ball enclosing k points, and many others. In several cases we get the first known linear time approximation algorithm for a given problem, where our approximation ratio matches that of previous work. The second topic we investigate is contour trees, a fundamental structure in computational topology. Contour trees give a compact summary of the evolution of level sets on a mesh, and are typically used on massive data sets. Previous algorithms for computing contour trees took Θ(n log n) time and were worst-case optimal. Here we provide an algorithm whose running time lies between Θ(nα(n)) and Θ(n log n), and varies depending on the shape of the tree, where α(n) is the inverse Ackermann function. In particular, this is the first algorithm with O(nα(n)) running time on instances with balanced contour trees. Our algorithmic results are complemented by lower bounds indicating that, up to a factor of α(n), on all instance types our algorithm performs optimally. For the final topic, we consider the descriptive complexity of weighted Voronoi diagrams. Such diagrams have quadratic (or higher) worst-case complexity, however, as was the case for contour trees, here we push beyond worst-case analysis. A new diagram, called the candidate diagram, is introduced, which allows us to bound the complexity of weighted Voronoi diagrams arising from a particular probabilistic input model. Specifically, we assume weights are randomly permuted among fixed Voronoi sites, an assumption which is weaker than the more typical sampled locations assumption. Under this assumption, the expected complexity is shown to be near linear

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

Optimization problems in network connectivity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 115-120).Besides being one of the principal driving forces behind research in algorithmic theory for more than five decades, network optimization has assumed increased significance in recent times with the advent and widespread use of a variety of large-scale real-life networks. The primary goal of such networks is to connect vertices (representing a variety of real-life entities) in a robust and inexpensive manner, and to store and retrieve such connectivity information efficiently. In this thesis, we present efficient algorithms aimed at achieving these broad goals. The main results presented in this thesis are as follows. -- Cactus Construction. We give a near-linear time Monte Carlo algorithm for constructing a cactus representation of all the minimum cuts in an undirected graph. -- Cut Sparsification. A cut sparsifier of an undirected graph is a sparse graph on the same set of vertices that preserves its cut values up to small errors. We give new combinatorial and algorithmic results for constructing cut sparsifiers. -- Online Steiner Tree. Given an undirected graph as input, the goal of the Steiner tree problem is to select its minimum cost subgraph that connects a designated subset of vertices. We give the first online algorithm for the Steiner tree problem that has a poly-logarithmic competitive ratio when the input graph has both node and edge costs. -- Network Activation Problems. In the design of real-life wireless networks, a typical objective is to select one among a possible set of parameter values at each node such that the set of activated links satisfy some desired connectivity properties. We formalize this as the network activation model, and give approximation algorithms for various fundamental network design problems in this model.by Debmalya Panigrahi.Ph.D

Coalition structure generation over graphs

We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Railway Timetable Optimization

In this cumulative dissertation, we study several aspects of railway timetable optimization. The first contributions cover Practical Applications of Automatic Railway Timetabling. In particular, for the problem of simultaneously scheduling all freight trains in Germany such that there are no conflicts between them, we propose a novel column generation approach. Each train can choose from an iteratively growing set of possible routes and times, so called slots. For the task of choosing maximally many slots without conflicts, we present and apply the heuristic algorithm Conflict Resolving (CR). With these two methods, we are able to schedule more than 5000 trains simultaneously, exceeding the scopes of other studies. A second practical application that we study is measuring the capacity increase in the railway network when equipping freight trains with electro-pneumatic brakes and middle buffer couplings. Methodically, we propose to explicitly construct as many slots as possible for such trains and measure the capacity as the number of constructed slots. Furthermore, we contribute to the field of Algorithms and Computability in Timetable Generation. We present two heuristic solution algorithms for the Maximum Satisfiability Problem (MaxSAT). In the literature, it has been proposed to encode different NP-complete problems that occur in railway timetabling in MaxSAT. In numerical experiments, we prove that our algorithms are competitive to state-of-the-art MaxSAT solvers. Moreover, we study the parameterized complexity status of periodic scheduling and give proofs that the problem is NP-complete for input graphs of bounded treewidth, branchwidth and carvingwidth. Finally, we propose a framework for analyzing Delay Propagation in Railway Networks. More precisely, we develop delay transmission rules based on different correlation measures that can be derived from historical operations data. What is more, we apply SHAP values from Explainable AI to the problem of discerning primary delays that occur stochastically in the operations, to secondary follow-up delays. Transmission rules that are derived from the secondary delays indicate where timetable adjustments are needed. In our last contribution in this field, we apply such adjustment rules for black-box optimization of timetables in a simulation environment

Fully Dynamic MIS in Uniformly Sparse Graphs

We consider the problem of maintaining a maximal independent set (MIS) in a dynamic graph subject to edge insertions and deletions. Recently, Assadi, Onak, Schieber and Solomon (STOC 2018) showed that an MIS can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this paper we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity alpha, the amortized update time of our algorithm is O(alpha^2 * log^2 n), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs as well as some classes of "real world" graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m^{3/8 - epsilon}, for any constant epsilon > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m^{1/2}

Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing

Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM