This article presents a new search algorithm for the NP-hard problem of optimizing functions of binary variables that decompose according to a graphical model. It can be applied to models of any order and structure. The main novelty is a technique to constrain the search space based on the topology of the model. When pursued to the full search depth, the algorithm is guaranteed to converge to a global optimum, passing through a series of monotonously improving local optima that are guaranteed to be optimal within a given and increasing Hamming distance. For a search depth of 1, it specializes to Iterated Conditional Modes. Between these extremes, a useful tradeoff between approximation quality and runtime is established. Experiments on models derived from both illustrative and real problems show that approximations found with limited search depth match or improve those obtained by state-of-the-art methods based on message passing and linear programming.
Naive attempts to generalize ICM and Block-ICM to optimize over subgraphs of size @math would consider all sequences of @math connected variables and ignore the fact that many of these sequences represent the same set. This causes substantial problems because the redundancy is large, as we show in . The Lazy Flipper avoids this redundancy, at the cost of storing one unique representative for each subset. Compared to randomized algorithms that sample from the set of subgraphs @cite_22 @cite_24 @cite_33 , this is a memory intensive approach. Up to 8 GB of RAM are required for the optimizations shown in . Now that servers with much larger RAM are available, it has become a practical option.
@cite_24 @cite_22 @cite_33
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In this paper, we study the problem of subgraph matching that extracts all subgraph isomorphic embeddings of a query graph q in a large data graph G. The existing algorithms for subgraph matching follow Ullmann's backtracking approach; that is, iteratively map query vertices to data vertices by following a matching order of query vertices. It has been shown that the matching order of query vertices is a very important aspect to the efficiency of a subgraph matching algorithm. Recently, many advanced techniques, such as enforcing connectivity and merging similar vertices in query or data graphs, have been proposed to provide an effective matching order with the aim to reduce unpromising intermediate results especially the ones caused by redundant Cartesian products. In this paper, for the first time we address the issue of unpromising results by Cartesian products from "dissimilar" vertices. We propose a new framework by postponing the Cartesian products based on the structure of a query to minimize the redundant Cartesian products. Our second contribution is proposing a new path-based auxiliary data structure, with the size O(|E(G)| x |V(q)|), to generate a matching order and conduct subgraph matching, which significantly reduces the exponential size O(|V(G)||V(q)|-1) of the existing path-based auxiliary data structure, where V (G) and E (G) are the vertex and edge sets of a data graph G, respectively, and V (q) is the vertex set of a query @math . Extensive empirical studies on real and synthetic graphs demonstrate that our techniques outperform the state-of-the-art algorithms by up to @math orders of magnitude. Motivated by applications in which the data may be formulated as a matrix, we consider algorithms for several common linear algebra problems. These algorithms make more efficient use of computational resources, such as the computation time, random access memory (RAM), and the number of passes over the data, than do previously known algorithms for these problems. In this paper, we devise two algorithms for the matrix multiplication problem. Suppose @math and @math (which are @math and @math , respectively) are the two input matrices. In our main algorithm, we perform @math independent trials, where in each trial we randomly sample an element of @math with an appropriate probability distribution @math on @math . We form an @math matrix @math consisting of the sampled columns of @math , each scaled appropriately, and we form a @math matrix @math using the corresponding rows of @math , again scaled appropriately. The choice of @math and the column and row scaling are crucial features of the algorithm. When these are chosen judiciously, we show that @math is a good approximation to @math . More precisely, we show that @math where @math denotes the Frobenius norm, i.e., @math . This algorithm can be implemented without storing the matrices @math and @math in RAM, provided it can make two passes over the matrices stored in external memory and use @math additional RAM to construct @math and @math . We then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm. In addition, we present a model (the pass-efficient model) in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets. In this model, the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm. The input matrices may be presented in any order of the entries (and not just row or column order), as is the case in many applications where, e.g., the data has been written in by multiple agents. In addition, the input matrices may be presented in a sparse representation, where only the nonzero entries are written. We present an @math -approximation algorithm for the problem of finding a @math -vertex connected spanning subgraph of minimum cost, where @math is the number of vertices in an input graph, and @math is a connectivity requirement. Our algorithm is the first that achieves a polylogarithmic approximation ratio for all values of @math and @math , and it works for both directed and undirected graphs. As in previous works, we use the Frank--Tardos algorithm for finding @math -outconnected subgraphs as a subroutine. However, with our structural lemmas, we are able to show that we need only partial solutions returned by the Frank--Tardos algorithm; thus, we can avoid paying the whole cost of an optimal solution every time the algorithm is applied. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . The Connectivity Server is a special-purpose database whose schema models the Web as a graph: a set of nodes (URL) connected by directed edges (hyperlinks). The Link Database provides fast access to the hyperlinks. To support easy implementation of a wide range of graph algorithms we have found it important to fit the Link Database into RAM. In the first version of the Link Database, we achieved this fit by using machines with lots of memory (8 GB), and storing each hyperlink in 32 bits. However, this approach was limited to roughly 100 million Web pages. This paper presents techniques to compress the links to accommodate larger graphs. Our techniques combine well-known compression methods with methods that depend on the properties of the Web graph. The first compression technique takes advantage of the fact that most hyperlinks on most Web pages point to other pages on the same host as the page itself. The second technique takes advantage of the fact that many pages on the same host share hyperlinks, that is, they tend to point to a common set of pages. Together, these techniques reduce space requirements to under 6 bits per link. While (de)compression adds latency to the hyperlink access time, we can still compute the strongly connected components of a 6 billion-edge graph in 22 minutes and run applications such as Kleinberg's HITS in real time. This paper describes our techniques for compressing the Link Database, and provides performance numbers for compression ratios and decompression speed. A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. The minimum cost subset k-connected subgraph problem is a cornerstone problem in the area of network design with vertex connectivity requirements. In this problem, we are given a graph G=(V,E) with costs on edges and a set of terminals T⊆V. The goal is to find a minimum cost subgraph such that every pair of terminals are connected by k openly (vertex) disjoint paths. In this paper, we present an approximation algorithm for the subset k-connected subgraph problem which improves on the previous best approximation guarantee of O(k2logk) by Nutov (ACM Trans. Algorithms 9(1):1, 2012). Our approximation guarantee, ?(|T|), depends upon the number of terminals: @math (|T|) = O(k ^2 k) & if 2k |T| k. This suggests that the hardest instances of the problem are when |T|?k. A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets @math of @math , satisfying @math , find a graph G with vertex set X and the minimum number of edges such that for any i, the subgraph induced by @math has a connected component containing @math .Two other problems related to this one are the following ones. (2) Given a set X and subsets @math such that @math , find a graph G with vertex set X and the minimum number of edges such that for any i the subgraph @math induced by @math in G is connected. (3) Given a set X and subsets @math such that @math , find a graph G with vertex set X, find a graph G with vertex set X and the minimum number of edges such that for any subset I of @math , the subgraph induced by @math is co... The problem of cache enabled private information retrieval (PIR) is considered in which a user wishes to privately retrieve one out of @math messages, each of size @math bits from @math distributed databases. The user has a local cache of storage @math bits which can be used to store any function of the @math messages. The main contribution of this work is the exact characterization of the capacity of cache aided PIR as a function of the storage parameter @math . In particular, for a given cache storage parameter @math , the information-theoretically optimal download cost @math (or the inverse of capacity) is shown to be equal to @math . Special cases of this result correspond to the settings when @math , for which the optimal download cost was shown by Sun and Jafar to be @math , and the case when @math , i.e., cache size is large enough to store all messages locally, for which the optimal download cost is @math . The intermediate points @math can be readily achieved through a simple memory-sharing based PIR scheme. The key technical contribution of this work is the converse, i.e., a lower bound on the download cost as a function of storage @math which shows that memory sharing is information-theoretically optimal. We present an iterative approximate solution to the multidimensional assignment problem under general cost functions. The method maintains a feasible solution at every step, and is guaranteed to converge. It is similar to the iterated conditional modes (ICM) algorithm, but applied at each step to a block of variables representing correspondences between two adjacent frames, with the optimal conditional mode being calculated exactly as the solution to a two-frame linear assignment problem. Experiments with ground-truthed trajectory data show that the method outperforms both network-flow data association and greedy recursive filtering using a constant velocity motion model. We develop a new technique for proving cell-probe lower bounds for static data structures. Previous lower bounds used a reduction to communication games, which was known not to be tight by counting arguments. We give the first lower bound for an explicit problem which breaks this communication complexity barrier. In addition, our bounds give the first separation between polynomial and near linear space. Such a separation is inherently impossible by communication complexity. Using our lower bound technique and new upper bound constructions, we obtain tight bounds for searching predecessors among a static set of integers. Given a set Y of n integers of l bits each, the goal is to efficiently find predecessor(x) = max y in Y | y <= x , by representing Y on a RAM using space S. In external memory, it follows that the optimal strategy is to use either standard B-trees, or a RAM algorithm ignoring the larger block size. In the important case of l = c*lg n, for c>1 (i.e. polynomial universes), and near linear space (such as S = n*poly(lg n)), the optimal search time is Theta(lg l). Thus, our lower bound implies the surprising conclusion that van Emde Boas' classic data structure from [FOCS'75] is optimal in this case. Note that for space n^ 1+eps , a running time of O(lg l lglg l) was given by Beame and Fich [STOC'99]. Frequent subgraph mining is an active research topic in the data mining community. A graph is a general model to represent data and has been used in many domains like cheminformatics and bioinformatics. Mining patterns from graph databases is challenging since graph related operations, such as subgraph testing, generally have higher time complexity than the corresponding operations on itemsets, sequences, and trees, which have been studied extensively. We propose a novel frequent subgraph mining algorithm: FFSM, which employs a vertical search scheme within an algebraic graph framework we have developed to reduce the number of redundant candidates proposed. Our empirical study on synthetic and real datasets demonstrates that FFSM achieves a substantial performance gain over the current start-of-the-art subgraph mining algorithm gSpan. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . The binary symmetric stochastic block model deals with a random graph of @math vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability @math within clusters and @math across clusters. In the asymptotic regime of @math and @math for fixed @math and @math , we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe14 . Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to @math . Given an undirected graph G = (V ,E ), the density of a subgraph on vertex set S is defined as @math , where E (S ) is the set of edges in the subgraph induced by nodes in S . Finding subgraphs of maximum density is a very well studied problem. One can also generalize this notion to directed graphs. For a directed graph one notion of density given by Kannan and Vinay [12] is as follows: given subsets S and T of vertices, the density of the subgraph is @math , where E (S ,T ) is the set of edges going from S to T . Without any size constraints, a subgraph of maximum density can be found in polynomial time. When we require the subgraph to have a specified size, the problem of finding a maximum density subgraph becomes NP -hard. In this paper we focus on developing fast polynomial time algorithms for several variations of dense subgraph problems for both directed and undirected graphs. When there is no size bound, we extend the flow based technique for obtaining a densest subgraph in directed graphs and also give a linear time 2-approximation algorithm for it. When a size lower bound is specified for both directed and undirected cases, we show that the problem is NP-complete and give fast algorithms to find subgraphs within a factor 2 of the optimum density. We also show that solving the densest subgraph problem with an upper bound on size is as hard as solving the problem with an exact size constraint, within a constant factor. (MATH) Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity: design a data structure for an undirected graph @math and a subset of vertices @math , to support insertions and deletions in @math and connectivity queries (are two vertices connected @?) in the subgraph induced by @math . We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using an elegant combination of several simple ideas. Our method requires linear space, @math amortized update time, and @math query time, where @math is the matrix multiplication exponent and @math hides polylogarithmic factors. Oblivious RAM is a useful primitive that allows a client to hide its data access patterns from an untrusted server in storage outsourcing applications. Until recently, most prior works on Oblivious RAM aim to optimize its amortized cost, while suffering from linear or even higher worst-case cost. Such poor worst-case behavior renders these schemes impractical in realistic settings, since a data access request can occasionally be blocked waiting for an unreasonably large number of operations to complete. This paper proposes novel Oblivious RAM constructions that achieves poly-logarithmic worst-case cost, while consuming constant client-side storage. To achieve the desired worst-case asymptotic performance, we propose a novel technique in which we organize the O-RAM storage into a binary tree over data buckets, while moving data blocks obliviously along tree edges. The block coordinate descent (BCD) method is widely used for minimizing a continuous function @math of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of @math which are either locally tight upper bounds of @math or strictly convex local approximations of @math . The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable or nonconvex. Our results unify and extend the existing convergence results ... Previous research indicates that during the past 20 years, the highest-quality work has been published in an increasingly diverse and larger group of journals. In this article, we examine whether this diversification has also affected the handful of elite journals that are traditionally considered to be the best. We examine citation patterns during the past 40 years of seven long-standing traditionally elite journals and six journals that have been increasing in importance during the past 20 years. To be among the top 5 or 1 cited papers, papers now need about twice as many citations as they did 40 years ago. Since the late 1980s and early 1990s, elite journals have been publishing a decreasing proportion of these top-cited papers. This also applies to the two journals that are typically considered as the top venues and often used as bibliometric indicators of "excellence": Science and Nature. On the other hand, several new and established journals are publishing an increasing proportion of the most-cited papers. These changes bring new challenges and opportunities for all parties. Journals can enact policies to increase or maintain their relative position in the journal hierarchy. Researchers now have the option to publish in more diverse venues knowing that their work can still reach the same audiences. Finally, evaluators and administrators need to know that although there will always be a certain prestige associated with publishing in "elite" journals, journal hierarchies are in constant flux. The NP-hard Subset Interconnection Design problem, also known as Minimum Topic-Connected Overlay, is motivated by numerous applications including the design of scalable overlay networks and vacuum systems. It has as input a finite set @math and a collection of subsets @math , and asks for a minimum-cardinality edge set @math such that for the graph @math all induced subgraphs @math are connected. We study Subset Interconnection Design in the context of polynomial-time data reduction rules that preserve the possibility of constructing optimal solutions. Our contribution is threefold: First, we show the incorrectness of earlier polynomial-time data reduction rules. Second, we show linear-time solvability in case of a constant number @math of subsets, implying fixed-parameter tractability for the parameter @math . Third, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. To achieve our results, we elaborate o...