39,905 research outputs found

    Planted Models for the Densest k-Subgraph Problem

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    Given an undirected graph G, the Densest k-subgraph problem (DkS) asks to compute a set S ? V of cardinality |S| ? k such that the weight of edges inside S is maximized. This is a fundamental NP-hard problem whose approximability, inspite of many decades of research, is yet to be settled. The current best known approximation algorithm due to Bhaskara et al. (2010) computes a ?(n^{1/4 + ?}) approximation in time n^{?(1/?)}, for any ? > 0. We ask what are some "easier" instances of this problem? We propose some natural semi-random models of instances with a planted dense subgraph, and study approximation algorithms for computing the densest subgraph in them. These models are inspired by the semi-random models of instances studied for various other graph problems such as the independent set problem, graph partitioning problems etc. For a large range of parameters of these models, we get significantly better approximation factors for the Densest k-subgraph problem. Moreover, our algorithm recovers a large part of the planted solution

    Constant time algorithms in sparse graph model

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 87-91).We focus on constant-time algorithms for graph problems in bounded degree model. We introduce several techniques to design constant-time approximation algorithms for problems such as Vertex Cover, Maximum Matching, Maximum Weighted Matching, Maximum Independent Set and Set Cover. Some of our techniques can also be applied to design constant-time testers for minor-closed properties. In Chapter 1, we show how to construct a simple oracle that provides query access to a fixed Maximal Independent Set (MIS) of the input graph. More specifically, the oracle gives answers to queries of the form "Is v in the MIS?" for any vertex v in the graph. The oracle runs in constant-time, i.e., the running time for the oracle to answer a single query, is independent to the size of the input graph. Combining this oracle with a simple sampling scheme immediately implies an approximation algorithm for size of the minimum vertex cover. The second technique, called oracle hierarchy, transforms classical approximation algorithms into constant-time algorithms that approximate the size of the optimal solution. The technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. In the transformation, oracle hierarchy uses the MIS oracle to simulates each phase. The problems amenable to these techniques include Maximum Matching, Maximum Weight Matching, Set Cover, and Minimum Dominating Set. For example, for Maximum Matching, we give the first constant-time algorithm that for the class of graphs of degree bounded by d, computes the maximum matching size to within en, for any e > 0, where n is the number of vertices in the graph. The running time of the algorithm is independent of n, and only depends on d and e. In Chapter 2, we introduce a new tool called partitioning oracle which provides query access to a fixed partition of the input graph. In particular, the oracle gives answers to queries of the form "Which part in the fixed partition contains v?" for any vertex v in the graph. We develop methods for constructing a partitioning oracle for any class of bounded-degree graphs with an excluded minor. For any e > 0, our partitioning oracle provides query access to a fixed partition of the input constant-degree minor-free graph, in which every part has size 0(1/ 2 ), and the number of edges removed is at most en. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance: " We give constant-time approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor. * We show a simple proof that any minor-closed graph property is testable in constant time in the bounded degree model. Finally, in Chapter 3, we construct a more efficient partitioning oracle for graphs with constant treewidth. Although the partitioning oracle in Chapter 2 runs in time independent of the size of the input graph, it has to make 2POlY(1/E)) queries to the input graph to answer a query about the partition. Our new partitioning oracle improves this query complexity to poly(1/E) for graphs with constant treewidth. The new oracle can be used to test constant treewidth in poly(1/E) time in the bounded-degree model. Another application is a poly(1/E)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive en in bounded treewidth graphs.by Huy Ngoc Nguyen.Ph.D

    Approximate Computation and Implicit Regularization for Very Large-scale Data Analysis

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    Database theory and database practice are typically the domain of computer scientists who adopt what may be termed an algorithmic perspective on their data. This perspective is very different than the more statistical perspective adopted by statisticians, scientific computers, machine learners, and other who work on what may be broadly termed statistical data analysis. In this article, I will address fundamental aspects of this algorithmic-statistical disconnect, with an eye to bridging the gap between these two very different approaches. A concept that lies at the heart of this disconnect is that of statistical regularization, a notion that has to do with how robust is the output of an algorithm to the noise properties of the input data. Although it is nearly completely absent from computer science, which historically has taken the input data as given and modeled algorithms discretely, regularization in one form or another is central to nearly every application domain that applies algorithms to noisy data. By using several case studies, I will illustrate, both theoretically and empirically, the nonobvious fact that approximate computation, in and of itself, can implicitly lead to statistical regularization. This and other recent work suggests that, by exploiting in a more principled way the statistical properties implicit in worst-case algorithms, one can in many cases satisfy the bicriteria of having algorithms that are scalable to very large-scale databases and that also have good inferential or predictive properties.Comment: To appear in the Proceedings of the 2012 ACM Symposium on Principles of Database Systems (PODS 2012

    Algorithmic and Statistical Perspectives on Large-Scale Data Analysis

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    In recent years, ideas from statistics and scientific computing have begun to interact in increasingly sophisticated and fruitful ways with ideas from computer science and the theory of algorithms to aid in the development of improved worst-case algorithms that are useful for large-scale scientific and Internet data analysis problems. In this chapter, I will describe two recent examples---one having to do with selecting good columns or features from a (DNA Single Nucleotide Polymorphism) data matrix, and the other having to do with selecting good clusters or communities from a data graph (representing a social or information network)---that drew on ideas from both areas and that may serve as a model for exploiting complementary algorithmic and statistical perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors, "Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201
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