Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retrieval application, and which should also be useful in other applications. The goal is to efficiently maintain clusters of small diameter as new points are inserted. We analyze several natural greedy algorithms and demonstrate that they perform poorly. We propose new deterministic and randomized incremental clustering algorithms which have a provably good performance. We complement our positive results with lower bounds on the performance of incremental algorithms. Finally, we consider the dual clustering problem where the clusters are of fixed diameter, and the goal is to minimize the number of clusters. 1 Introduction We consider the following problem: as a sequence of points from a metric...

In this paper, we study approximation algorithms for several NP-hard facility location problems. We prove that a simple local search heuristic yields polynomial-time constant-factor approximation bounds for the metric versions of the uncapacitated k-median problem and the uncapacitated facility location problem. (For the k-median problem, our algorithms require a constant-factor blowup in the parameter k.) This local search heuristic was rst proposed several decades ago, and has been shown to exhibit good practical performance in empirical studies. We also extend the above results to obtain constant-factor approximation bounds for the metric versions of capacitated k-median and facility location problems.

In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixed-parameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that k-Step Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph k-Coloring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.

We consider the problem of scheduling wireless data in systems such as 802.16 (WIMAX). Each scheduling decision involves constructing a frame of one or more time slots. Within each time slot multiple carriers must be assigned to users. The important aspect of our problem is that a scheduler knows the channel rates across all users and all carriers whenever a scheduling decision is made. Hence there is no need to treat each carrier in complete isolation. This gives a potential for enhancing performance by allocating multiple carriers simultaneously. We analyze this problem in a situation where finite queues are fed by a data arrival process. We generalize the wellknown MaxWeight algorithm for the single-carrier setting to accommodate a number of natural optimization problems in the multi-carrier setting. We state the hardness of these problems and present simple algorithmic solutions with provable performance bounds. We also validate our algorithms via numerical examples.

We develop a framework for obtaining Fully Polynomial Time Approximation Schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. Using our framework, we give the first FPTASs for several NP-hard problems in various fields of research such as knapsack-related problems, logistics, operations management, economics, and mathematical finance.

An important problem in VLSI design is distributing a clock signal to synchronous elements in a VLSI circuit so that the signal arrives at all elements simultaneously. The signal is distributed by means of a clock routing tree rooted at a global clock source. The difference in length between the longest and shortest root-leaf path is called the skew of the tree. The problem is to construct a clock tree with zero skew (to achieve synchronicity) and minimal sum of edge lengths (so that circuit area and clock tree capacitance are minimized). We give the first constant-factor approximation algorithms for this problem and its variants that arise in the VLSI context. For the zero skew problem in general metric spaces, we give an approximation algorithm with a performance guarantee of 2e. For the L 1 version on the plane, we give an (8/ ln 2)- approximation algorithm. 1 Introduction. A fundamental problem in VLSI design is clock routing, i.e., distributing a clock signal to synchro...

We present an algorithm that generates all (inclusion-wise) minimal feedback vertex sets of a directed graph G = (V; E). The feedback vertex sets of G are generated with a polynomial delay of O \Gamma jV j 2 (jV j + jEj) \Delta . Variants of the algorithm generate all minimal solutions for the undirected case and the directed feedback arc set problem, both with a polynomial delay of O \Gamma jV j jEj (jV j + jEj) \Delta . 1 Introduction Generating all admissible configurations is a well-examined problem for many combinatorial problems. Typically, solutions are subsets of a finite set, and the set of solutions is monotone, i.e. the supersets of admissible solutions are also admissible. This is the case for the feedback problems that we examine, and thus the enumeration of all minimal admissible solutions provides a generic nonredundant description of the solution space. For a directed graph, a feedback vertex set is a subset of its vertices that contains at least one vertex ...

We present an alg orithm thatg enerates all (inclusion-wise) minimal feedback vertex sets of a directedg raph G =(V,E). The feedback vertex sets of G areg enerated with a polynomial delay of O # |V | 2 (|V | + |E|) # . We further show that the underlying technique can be tailored tog enerate all minimal solutions for the undirected case and the directed feedback arc set problem, both with a polynomial delay of O # |V ||E| (|V | + |E|) # . Finally we prove that computing the number of minimal feedback arc sets is #P-hard.

Abstract. We consider the problem of covering and packing subsets of δ-hyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δhyperbolic graph G and a positive number R, let γ(S, R) be the minimum number of balls of radius R covering S. It is known that computing γ(S, R) or approximating this number within a constant factor is hard even for 2-hyperbolic graphs. In this paper, using a primal-dual approach, we show how to construct in polynomial time a covering of S with at most γ(S, R) balls of (slightly larger) radius R + δ. This result is established in the general framework of δ-hyperbolic geodesic metric spaces and is extended to some other set families derived from balls. This covering algorithm is used to design better than in general case approximation algorithms for the augmentation problem of δ-hyperbolic graphs with diameter constraints and slackness δ and for the k-center problem in δ-hyperbolic graphs. 1

We study the problem of sampling from a large genetic mapping population in which all individuals have identical pedigrees. We show that samples obtained from large populations, selected on the basis of limited genetic data, are better suited for use in high-density mapping experiments than random samples of the same size. We model the problem of choosing a mapping sample as a discrete stochastic optimization problem, related to existing clustering problems, and study various heuristics for the problem, including some randomized rounding algorithms. Experiments on both simulated data and ten data sets from biological populations show that these heuristics perform very well in practice despite the problem being NP-hard to approximate to within any constant. Our proposals oer the possibility of higher resolution, less expensive genetic maps.