The development of novel platforms and techniques for emerging "Big Data" applications requires the availability of real-life datasets for data-driven experiments, which are however out of reach for academic research in most cases as they are typically proprietary. A possible solution is to use synthesized datasets that reflect patterns of real ones in order to ensure high quality experimental findings. A first step in this direction is to use inverse mining techniques such as inverse frequent itemset mining (IFM) that consists of generating a transactional database satisfying given support constraints on the itemsets in an input set, that are typically the frequent ones. This paper introduces an extension of IFM, called many-sorted IFM, where the schemes for the datasets to be generated are those typical of Big Tables as required in emerging big data applications, e.g., social network analytics.
A reformulation of in terms of frequencies instead of supports has been introduced in @cite_3 @cite_6 with the name @math . The two problems are equivalent and have been shown to be in @math and @math -hard. The basic version of the frequency formulation, called , does not fix the number @math of transaction in a feasible database -- the corresponding decision problem has been proved to be @math -complete. A further variant of the problem has been introduced in @cite_3 @cite_6 with the name @math : all itemsets may occur as transactions in @math at most a fixed number of times ( @math ). This problem is in @math and @math -hard.
@cite_6 @cite_3
2026815944 2007278728
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 . We consider the @math -Center problem and some generalizations. For @math -Center a set of @math center vertices needs to be found in a graph @math with edge lengths, such that the distance from any vertex of @math to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter @math and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameter @math , but implies that such an algorithm must have at least doubly exponential running time in @math if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of @math by combining the parameters @math and @math : we develop a fixed-parameter @math -approximation with running time @math . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter @math -approximations for only the parameter @math . We also provide similar fixed-parameter approximations for the weighted @math -Center and @math -Partition problems, which generalize @math -Center. The centralized coded caching scheme is a technique proposed by Maddah-Ali and Niesen as a method to reduce the network burden in peak times in a wireless network system. reformulate the problem as designing a corresponding placement delivery array and propose two new schemes from this perspective. These schemes significantly reduce the rate compared with the uncoded caching schemes. However, to implement these schemes, each file should be cut into @math pieces, where @math grows exponentially with the number of users @math . Such a constraint is obviously infeasible in the practical setting, especially when @math is large. Thus, it is desirable to design caching schemes with constant rate @math (independent of @math ) as well as smaller @math . In this paper, we view the centralized coded caching problem in a hypergraph perspective and show that designing a feasible placement delivery array is equivalent to constructing a linear and (6,3)-free 3-uniform 3-partite hypergraph. Several new results and constructions arise from our novel point of view. First, by using the famous (6,3)-theorem in extremal graph theory, we show that constant rate placement delivery arrays with @math growing linearly with @math do not exist. Second, we present two infinite classes of placement delivery arrays to show that constant rate caching schemes with @math growing sub-exponentially with @math do exist. We consider the k-Center problem and some generalizations. For k-Center a set of kcenter vertices needs to be found in a graph G with edge lengths, such that the distance from any vertex ofi¾?G to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [ICALP 2011]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter k, and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameteri¾?h, but implies that such an algorithm must have at least doubly exponential running time in h if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor ofi¾?2 by combining the parameters k andi¾?h: we develop a fixed-parameter 3 2-approximation with running time @math . We also provide similar fixed-parameter approximations for the weightedk-Center and @math -Partition problems, which generalize k-Center. Given a directed graph @math and a list @math of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of @math that contains a directed @math path for every @math . The special case Directed Steiner Tree (when we ask for paths from a root @math to terminals @math ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every @math to every other @math ) is known to be W[1]-hard. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if @math is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list @math of requests form a directed graph that is a member of @math . Our main result is a complete characterization of the classes @math resulting in fixed-parameter tractable special cases: we show that if every pattern in @math has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable @math not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], @math -Root Steiner Tree is FPT for constant @math [Such 'y, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [, SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before. In this paper, we study the dominance relation under a stochastic setting. Let @math be a set of @math colored stochastic points in @math , each of which is associated with an existence probability. We investigate the problem of computing the probability that a realization of @math contains inter-color dominances, which we call the (CSD) problem. We propose the first algorithm to solve the CSD problem for @math in @math time. On the other hand, we prove that, for @math , even the CSD problem with a restricted color pattern is #P-hard. In addition, even if the existence probabilities are restricted to be @math , the problem remains #P-hard for @math . A simple FPRAS is then provided to approximate the desired probability in any dimension. We also study a variant of the CSD problem in which the dominance relation is considered with respect to not only the standard basis but any orthogonal basis of @math . Specifically, this variant, which we call the free-basis colored stochastic dominance (FBCSD) problem, considers the probability that a realization of @math contains inter-color dominances with respect to any orthogonal basis of @math . We show that the CSD problem is polynomial-time reducible to the FBCSD problem in the same dimension, which proves the #P-hardness of the latter for @math . Conversely, we reduce the FBCSD problem in @math to the CSD problem in @math , by which an @math time algorithm for the former is obtained. A Fixed-Parameter Tractable ( ) @math -approximation algorithm for a minimization (resp. maximization) parameterized problem @math is an FPT algorithm that, given an instance @math computes a solution of cost at most @math (resp. @math ) if a solution of cost at most (resp. at least) @math exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard Clique and W[2]-hard Set Cover problems, the natural question is whether we can get any -approximation. It is widely believed that both Clique and Set-Cover admit no FPT @math -approximation algorithm, for any increasing function @math . Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) eth-paturi and the Projection Games Conjecture (PGC) r3 , we make the first progress towards proving this conjecture by showing that 1. Under the ETH and PGC, there exist constants @math such that the Set Cover problem does not admit an FPT approximation algorithm with ratio @math in @math time, where @math is the size of the universe and @math is the number of sets. 2. Unless @math , for every @math there exists a constant @math such that Clique has no FPT cost approximation with ratio @math in @math time, where @math is the number of vertices in the graph. In the second part of the paper we consider various W[1]-hard problems such as , , Directed Steiner Network and . For all these problem we give polynomial time @math -approximation algorithms for some small function @math (the largest approximation ratio we give is @math ). We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W @math , for @math , that corresponds to Downey and Fellows's W-hierarchy [R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999] and we show that a few central W-completeness results for decision problems translate to #W-completeness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W @math f(k) n^c @math f: N N @math 2^ O(k) n^ 2.376 $ algorithm for finding a cycle or path of length k [N. Alon, R. Yuster, and U. Zwick, J. ACM, 42 (1995), pp. 844--856]. We consider the maximum independent set problem on graphs with maximum degree @math . We show that the integrality gap of the Lov 'asz @math -function based SDP is @math . This improves on the previous best result of @math , and almost matches the integrality gap of @math recently shown for stronger SDPs, namely those obtained using poly- @math levels of the @math semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of @math -free graphs for large values of @math . We also show how to obtain an algorithmic version of the above-mentioned @math -based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of @math matches the best unique-games-based hardness result up to lower-order poly- @math factors. Given two graphs @math and @math , the Subgraph Isomorphism problem asks if @math is isomorphic to a subgraph of @math . While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time @math using the color-coding technique of Alon, Yuster, and Zwick; (2) in time @math using Courcelle's Theorem; (3) in time @math using a result on first-order model checking by Frick and Grohe; or (4) in time @math for connected @math using the algorithm of Matou s ek and Thomas. Already this small sample of results shows that the way an algorithm can depend on the parameters is highly nontrivial and subtle. We develop a framework involving 10 relevant parameters for each of @math and @math (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time [ f_1(p_1,p_2,..., p_ ) n^ f_2(p_ +1 ,..., p_k) ] exist, where each of @math is one of the 10 parameters depending only on @math or @math . We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper. On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of @math gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when @math is a bounded-degree tree of constant pathwidth and @math is a planar graph of bounded pathwidth. Stable Marriage is a fundamental problem to both computer science and economics. Four well-known NP-hard optimization versions of this problem are the Sex-Equal Stable Marriage (SESM), Balanced Stable Marriage (BSM), max-Stable Marriage with Ties (max-SMT) and min-Stable Marriage with Ties (min-SMT) problems. In this paper, we analyze these problems from the viewpoint of Parameterized Complexity. We conduct the first study of these problems with respect to the parameter treewidth. First, we study the treewidth @math of the primal graph. We establish that all four problems are W[1]-hard. In particular, while it is easy to show that all four problems admit algorithms that run in time @math , we prove that all of these algorithms are likely to be essentially optimal. Next, we study the treewidth @math of the rotation digraph. In this context, the max-SMT and min-SMT are not defined. For both SESM and BSM, we design (non-trivial) algorithms that run in time @math . Then, for both SESM and BSM, we also prove that unless SETH is false, algorithms that run in time @math do not exist for any fixed @math . We thus present a comprehensive, complete picture of the behavior of central optimization versions of Stable Marriage with respect to treewidth. Finding the so-called characteristic numbers of the complex projective plane @math is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given @math and @math one has to find the number of degree @math genus @math curves that pass through a certain generic configuration of points and in the same time are tangent to certain generic configuration of lines. The total number of points and lines in these two configurations is @math so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when @math . Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we compute genus 0 characteristic numbers of @math in terms of open Hurwitz numbers. We study the optimal design problems where the goal is to choose a set of linear measurements to obtain the most accurate estimate of an unknown vector in @math dimensions. We study the @math -optimal design variant where the objective is to minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks, sparse least squares regression, feature selection for @math -means clustering, and matrix approximation. In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for @math -optimal design. Our main result is to obtain improved approximation algorithms for the @math -optimal design problem by introducing the proportional volume sampling algorithm. Our results nearly optimal bounds in the asymptotic regime when the number of measurements done, @math , is significantly more than the dimension @math . We also give first approximation algorithms when @math is small including when @math . The proportional volume-sampling algorithm also gives approximation algorithms for other optimal design objectives such as @math -optimal design and generalized ratio objective matching or improving previous best known results. Interestingly, we show that a similar guarantee cannot be obtained for the @math -optimal design problem. We also show that the @math -optimal design problem is NP-hard to approximate within a fixed constant when @math . We consider range queries that search for low-frequency elements (least frequent elements and @math ?-minorities) in arrays. An @math ?-minority of a query range has multiplicity no greater than an @math ? fraction of the elements in the range. Our data structure for the least frequent element range query problem requires @math O(n) space, @math O(n3 2) preprocessing time, and @math O(n) query time. A reduction from boolean matrix multiplication to this problem shows the hardness of simultaneous improvements in both preprocessing time and query time. Our data structure for the @math ?-minority range query problem requires @math O(n) space, supports queries in @math O(1 ?) time, and allows @math ? to be specified at query time. We introduce emph Normative Temporal Logic (acro ntl ), a logic for reasoning about normative systems. acro ntl is a generalisation of the well-known branching-time temporal logic acro ctl , in which the path quantifiers @math ( on all pathsldots'') and @math ( on some pathldots'') are replaced by the indexed deontic operators @math and @math , where for example @math means @math is obligatory in the context of normative system @math ''. After defining the logic, we give a sound and complete axiomatisation, and discuss the logic's relationship to standard deontic logics. We present a symbolic representation language for models and normative systems, and identify four different model checking problems, corresponding to whether or not a model is represented symbolically or explicitly, and whether or not we are given an interpretation for the normative systems named in formulae to be checked. We show that the complexity of model checking varies from acro p -complete up to acro exptime -hard for these variations. Given a simple undirected graph @math and a positive integer @math , the maximum vertex coverage problem (MVC) is the problem of finding a set @math of @math vertices of @math such that the number of edges having at least one endpoint in @math is as large as possible. The problem is NP-hard even in bipartite graphs, as shown in two recent papers [N. Apollonio and B. Simeone, Discrete Appl. Math., 165 (2014), pp. 37--48; G. Joret and A. Vetta, Reducing the Rank of a Matroid, preprint, arXiv:1211.4853v1 [cs.DS], 2012]. By exploiting the structure of the fractional optimal solutions of a linear programming formulation for the maximum coverage problem, we provide a @math -approximation algorithm for the problem. The algorithm immediately extends to the weighted version of MVC. In this work we analyze the sample complexity of classification by differentially private algorithms. Differential privacy is a strong and well-studied notion of privacy introduced by (2006) that ensures that the output of an algorithm leaks little information about the data point provided by any of the participating individuals. Sample complexity of private PAC and agnostic learning was studied in a number of prior works starting with (, 2008) but a number of basic questions still remain open, most notably whether learning with privacy requires more samples than learning without privacy. We show that the sample complexity of learning with (pure) differential privacy can be arbitrarily higher than the sample complexity of learning without the privacy constraint or the sample complexity of learning with approximate differential privacy. Our second contribution and the main tool is an equivalence between the sample complexity of (pure) differentially private learning of a concept class @math (or @math ) and the randomized one-way communication complexity of the evaluation problem for concepts from @math . Using this equivalence we prove the following bounds: 1. @math , where @math is the Littlestone's (1987) dimension characterizing the number of mistakes in the online-mistake-bound learning model. Known bounds on @math then imply that @math can be much higher than the VC-dimension of @math . 2. For any @math , there exists a class @math such that @math but @math . 3. For any @math , there exists a class @math such that the sample complexity of (pure) @math -differentially private PAC learning is @math but the sample complexity of the relaxed @math -differentially private PAC learning is @math . This resolves an open problem of (2013b). Given data stream @math of size @math of numbers from @math , the frequency of @math is defined as @math . The @math -th of @math is defined as @math . We consider the problem of approximating frequency moments in insertion-only streams for @math . For any constant @math we show an @math upper bound on the space complexity of the problem. Here @math is the iterative @math function. To simplify the presentation, we make the following assumptions: @math and @math are polynomially far; approximation error @math and parameter @math are constants. We observe a natural bijection between streams and special matrices. Our main technical contribution is a non-uniform sampling method on matrices. We call our method a ; it samples a heavy element (i.e., element @math with frequency @math ) with probability @math and gives approximation @math . In addition, the estimations never exceed the real values, that is @math for all @math . As a result, we reduce the space complexity of finding a heavy element to @math bits. We apply our method of recursive sketches and resolve the problem with @math bits. This paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion. The computational model is such that each node of the graph is occupied by a processor which has its own ID. Processors are restricted to collecting data from others which are at a distance at most t away from them in t time units, but are otherwise computationally unbounded. This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data.Three results are proved within this model: • A 3-coloring of an n-cycle requires time @math . This bound is tight, by previous work of Cole and Vishkin. • Any algorithm for coloring the d-regular tree of radius r which runs for time at most @math requires at least @math colors. • In an n-vertex graph of largest degree @math , an @math -coloring may be found in time @math . We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math .