We consider the following clustering problem: we have a complete graph on # vertices (items), where each edge ### ## is labeled either # or depending on whether # and # have been deemed to be similar or different. The goal is to produce a partition of the vertices (a clustering) that agrees as much as possible with the edge labels. That is, we want a clustering that maximizes the number of # edges within clusters, plus the number of edges between clusters (equivalently, minimizes the number of disagreements: the number of edges inside clusters plus the number of # edges between clusters). This formulation is motivated from a document clustering problem in which one has a pairwise similarity function # learned from past data, and the goal is to partition the current set of documents in a way that correlates with # as much as possible; it can also be viewed as a kind of "agnostic learning" problem. An interesting

We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimum k-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degree Ω(n), although our algorithms solve most of these problems so long as the average degree is Ω(n). Denseness for non-graph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.

We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the well-known LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.

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A polynomial time approximation scheme (PTAS) for an optimization problem A is an algorithm that given in input an instance of A and E> 0 find;,; (1 + E)-approximate solution in time that is polynomial for each fixed E. Typical running times are no(+) or 2” ’ n. While algorithms of the former kind tend to be impractical, the latter ones are more interesting. In several cases, the development of algorithms of the second type required considerably new, and sometimes harder, techniques. For some interesting problems, only n”(“E) approximation schemes are known. Under likely assumptions, we prove that for some problems (including natural ones) there cannot be approximation schemes running in time f ( l/n)nO(‘), no matter how fast function f grows. Our result relies on a connection with Parameterized Complexity Theory, and we show that this connection is necessary.

The recent explosion of interest in graph cut methods in computer vision naturally spawns the question: what energy functions can be minimized via graph cuts? This question was first attacked by two papers of Kolmogorov and Zabih [23, 24], in which they dealt with functions with pairwise and triplewise pixel interactions. In this work, we extend their results in two directions. First, we examine the case of k-wise pixel interactions; the results are derived from a purely algebraic approach. Second, we discuss the applicability of provably approximate algorithms. Both of these developments should help researchers best understand what can and cannot be achieved when designing graph cut based algorithms.

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E ′ P (G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. • For any fixed ɛ> 0 and any monotone property P, there is a deterministic algorithm, which given a graph G = (V, E) of size n, approximates E ′ P (G) in linear time O(|V | + |E|) to within an additive error of ɛn2. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E ′ P. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. 1. If there is a bipartite graph that does not satisfy P, then there is a δ> 0 for which it is

We propose a new formulation of the conceptual clustering problem where the goal is to explicitly output a collection of simple and meaningful conjunctions of attributes that define the clusters. The formulation differs from previous approaches since the clusters discovered may overlap and also may not cover all the points. In addition, a point may be assigned to a cluster description even if it only satisfies most, and not necessarily all, of the attributes in the conjunction. Connections between this conceptual clustering problem and the maximum edge biclique problem are made. Simple, randomized algorithms are given that discover a collection of approximate conjunctive cluster descriptions in sublinear time.

ABSTRACT: Recent work in the analysis of randomized approximation algorithms for NP-hard optimization problems has involved approximating the solution to a problem by the solution of a related subproblem of constant size, where the subproblem is constructed by sampling elements of the original problem uniformly at random. In light of interest in problems with a heterogeneous structure, for which uniform sampling might be expected to yield suboptimal results, we investigate the use of nonuniform sampling probabilities. We develop and analyze an algorithm which uses a novel sampling method to obtain improved bounds for approximating the Max-Cut of a graph. In particular, we show that by judicious choice of sampling probabilities one can obtain error bounds that are superior to the ones obtained by uniform sampling, both for unweighted and weighted versions of Max-Cut. Of at least as much interest as the results we derive are the techniques we use. The first technique is a method to compute a compressed approximate decomposition of a matrix as the product

We study three different de-randomization methods that are often applied to approximate combinatorial optimization problems. We analyze the conditional probabilities method in connection with randomized rounding for routing, packing and covering integer linear programming problems. We show extensions of such methods for nonindependent randomized rounding for the assignment problem. The second method, the so called random walks is exemplified with algorithms for dense instances of some NP problems. Another often used method is the bounded independence technique; we explicit this method for the sparsest cut and maximum concurrent flow problems. 1 Introduction Randomized algorithms are often used to solve or to approximate optimization problems related to network design. Theoretical and practical concern about the effective availability of a source of truly random bits, as well as the desire of improved reliability, motivate the search for deterministic versions of such algorithms. While...