Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although

Given a planar Rraph on n nodes with costs (weights) on its edges,- define;he distance between nodes i &d 2 as ’ the length of the shortest path between i and i. Consider this as &I instance of me & TSP. For any E> 6, our algorithm finds a salesman tour of total cost at most (1 + E) times optimal in time n”(llea). We also present a quasi-polynomial time algorithm for the Steiner version of this problem.

Localization is a critical issue in mobile robotics. If the robot does not know where it is, it, cannot effectively plan movements, locate objects, or reach goals. In this paper, we describe probabilistic self-localization techniques for mobile robots that are based on the principal of maximum-likelihood estimation. The basic method is to compare a map generated at the current robot position to a previously generated map of the environment to prohabilistically maximize the agreement between the maps. This method is able to operate in both indoor and outdoor environments using either discrete features or an occupancy grid to represent the world map. The map may be generated using any method to detect features in the robot's surroundings, including vision, sonar, a d laser range-finder. A global search of the pose space is performed that guarantees that the best position in a discretized pose space is found according to the probabilistic: map agreement measure. In addition, fitting the likelihood function with a parameterized smface allows both subpixel localization and uncertainty estimation to be performed. The application of these techniques in several experiments is described, including experimental localization results for the Sojourner Mars rover. 1

We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem.

NP-hard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NP-hard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), k-TSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), k-MST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum

This paper introduces a model for parallel computation, called the distributed random-access machine (DRAM), in which the communication requirements of parallel algorithms can be evaluated. A DRAM is an abstraction of a parallel computer in which memory accesses are implemented by routing messages through a communication network. A DRAM explicitly models the congestion of messages across cuts of the network. We introduce the notion of a conservative algorithm as one whose communication requirements at each step can be bounded by the congestion of pointers of the input data structure across cuts of a DRAM. We give a simple lemma that shows how to "shortcut" pointers in a data structure so that remote processors can communicate without causing undue congestion. We give O(lg n)-step, linear-processor, linear-space, conservative algorithms for a variety of problems on n- node trees, such as computing treewalk numberings, finding the separator of a tree, and evaluating all subexpressions ...

The class PTAS is defined to consist of all NP optimization problems that permit polynomial-time approximation schemes. This paper explores the possibility that a core of PTAS may be characterized through syntactic classes endowed with restrictions on the structure of the input instances. Recent work in approximability of NP-hard problems has led to the identification of a syntactic class called MAX SNP as the core of APX, the class of constant-factor approximable NP optimization problems. This has enhanced our understanding of these classes from both an algorithmic and a complexity-theoretic point of view. Our work is motivated by the hope that a similar understanding can be attained for PTAS. We argue that while the core of APX is the purely syntactic class MAX SNP, in the case of PTAS we must identify the core in terms of syntactic prescriptions for the problem definition augmented with structural restrictions on the input instances. Specifically, we propose such a unified framework...

In many models of optical routing, we are given a set of communication paths in a network, and we must assign a wavelength to each path so that paths sharing an edge receive different wavelengths. The goal is to assign as few wavelengths as possible, in order to make as efficient use as possible of the optical bandwidth. Much work in the area of optical networks has considered the use of wavelength converters: if a node of a network contains a wavelength converter, any path that passes through this node may change its wavelength. Having converters at some of the nodes can reduce the number of wavelengths required for routing, down to the following natural congestion bound: even with converters, we will always need at least as many wavelengths as the maximum number of paths sharing a single edge. Thus Wilfong and Winkler defined a set S of nodes in a network to be sufficient if, placing converters at the nodes in S, every set of paths can be routed with a number of wavelengths equal to its congestion bound. They showed that finding a sufficient set of minimum size is NP-complete, even in planar graphs. In this paper,

We demonstrate a new connection between fixed-parameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of so-called “bidimensional” problems to show that essentially all such problems have both subexponential fixed-parameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal problems, dominating set, edge dominating set, r-dominating set, diameter, connected dominating set, connected edge dominating set, and connected r-dominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two well-known problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the Lipton-Tarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.

Graph separation is a well-known tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many well-known NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p