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Property testing is a relaxation of classical decision problems which aims at distinguishing between functions having a predetermined property and functions being far from any function having the property. In this paper we present a novel framework for analyzing property testing algorithms. Our framework is based on a connection of property testing and a new class of problems which we call abstract combinatorial programs. We show that if the problem of testing a property can be reduced to an abstract combinatorial program of small dimension, then the property has an efficient tester. We apply our framework to a variety of problems. We present efficient property testing algorithms for geometric clustering problems, for the reversal distance problem, and for graph and hypergraph coloring problems. We also prove that, informally, any hereditary graph property can be efficiently tested if and only if it can be reduced to an abstract combinatorial program of small size. Our framework allows us to analyze all our testers in a unified way, and the obtained complexity bounds either match or improve the previously known bounds. Furthermore, even if the asymptotic complexity of the testers is not improved, the obtained proofs are significantly simpler than the previous ones. We believe that our framework will help to understand the structure of efficiently testable properties. [ABSTRACT FROM AUTHOR]

In this paper, we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. An instance of this problem consists of a set of k people and m indivisible goods. Each person has a known linear utility function over the set of goods which might be different from the utility functions of other people. The goal is to distribute the goods among the people and maximize the minimum utility received by them. The approximation ratio of our algorithm is Ω(1/√k log3 k). As a crucial part of our algorithm, we design and analyze an iterative method for rounding a fractional matching on a tree which might be of independent interest. We also provide better bounds when we are allowed to exclude a small fraction of the people from the problem. [ABSTRACT FROM AUTHOR]

Kernel-based linear-threshold algorithms, such as support vector machines and Perceptron-like algorithms, are among the best available techniques for solving pattern classification problems. In this paper, we describe an extension of the classical Perceptron algorithm, called second-order Perceptron, and analyze its performance within the mistake bound model of on-line learning. The bound achieved by our algorithm depends on the sensitivity to second-order data information and is the best known mistake bound for (efficient) kernel-based linear-threshold classifiers to date. This mistake bound, which strictly generalizes the well-known Perceptron bound, is expressed in terms of the eigenvalues of the empirical data correlation matrix and depends on a parameter controlling the sensitivity of the algorithm to the distribution of these eigenvalues. Since the optimal setting of this parameter is not known a priori, we also analyze two variants of the second-order Perceptron algorithm: one that adaptively sets the value of the parameter in terms of the number of mistakes made so far, and one that is parameterless, based on pseudoinverses. [ABSTRACT FROM AUTHOR]

We present a new analysis of the worst-case cost of path compression, which is an operation that is used in various well-known "union-find" algorithms. In contrast to previous analyses which are essentially based on bottom-up approaches, our method proceeds top-down, yielding recurrence relations from which the various bounds arise naturally. In particular the famous quasi-linear bound involving the inverse Ackermann function can be derived without having to introduce the Ackermann function itself. [ABSTRACT FROM AUTHOR]

Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call minimum conflict-free coloring (min-CF-coloring). In its general form, the input of the min-CF-coloring problem is a set system (X, S), where each S ϵ S is a subset of X. The output is a coloring x of the sets in S that satisfies the following constraint: for every x ϵ X there exists a color i and a unique set S ϵ S such that x ϵ S and x(S) = i. The goal is to minimize the number of colors used by the coloring x. Min-CF-coloring of general set systems is not easier than the classic graph coloring problem. However, in view of our motivation, we consider set systems induced by simple geometric regions in the plane. In particular, we study disks (both congruent and noncongruent), axis-parallel rectangles (with a constant ratio between the smallest and largest rectangle), regular hexagons (with a constant ratio between the smallest and largest hexagon), and general congruent centrally symmetric convex regions in the plane. In all cases we have coloring algorithms that use O(log n) colors (where n is the number of regions). Tightness is demonstrated by showing that even in the case of unit disks, θ(log n) colors may be necessary. For rectangles and hexagons we also obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also consider a dual problem of CF-coloring points with respect to sets. Given a set system (X, S), the goal in the dual problem is to color the elements in X with a minimum number of colors so that every set S ϵ S contains a point whose color appears only once in S. We show that O(log lXI) colors suffice for set systems in which X is a set of points in the plane and the sets are intersections of X with scaled translations of a convex region. [ABSTRACT FROM AUTHOR]

We consider suffix tree construction for situations with missing suffix links. Two examples of such situations are suffix trees for parameterized strings and suffix trees for two-dimensional arrays. These trees also have the property that the node degrees may be large. We add a new back- propagation component to MeCreight's algorithm and also give a high probability hashing scheme for large degrees. We show that these two features enable construction of suffix trees for general sit- uations with missing suffix links in O(n) time, with high probability. This gives the first randomized linear time algorithm for constructing suffix trees for parameterized strings. [ABSTRACT FROM AUTHOR]