Showing papers on "Disjoint sets published in 2009"
01 Jan 2009
TL;DR: A large body of research in supervised learning deals with the analysis of single-label data, where training examples are associated with a single label λ from a set of disjoint labels L, however, training examples in several application domains are often associated withA set of labels Y ⊆ L.
Abstract: A large body of research in supervised learning deals with the analysis of single-label data, where training examples are associated with a single label λ from a set of disjoint labels L. However, training examples in several application domains are often associated with a set of labels Y ⊆ L. Such data are called multi-label.
1,441 citations
TL;DR: In this article, a conformal field theory approach to entanglement entropy is presented, and the authors show how to apply these methods to the calculation of the entropy of a single interval and the generalization to different situations such as finite size, systems with boundaries and the case of several disjoint intervals.
Abstract: We review the conformal field theory approach to entanglement entropy. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries, and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.
1,006 citations
TL;DR: In this article, the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson) is studied.
Abstract: We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr ρAn for any integer n is calculated as the four-point function of twist fields of a particular type and the final result is expressed in a compact form in terms of the Riemann–Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.
381 citations
25 Oct 2009
TL;DR: In this article, the authors study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet, and show that no online algorithm can achieve an approximation ratio better than 0.632.
Abstract: We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of $1-{1\over e} \simeq 0.632$, a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the $1 - {1\over e}$ bound.Our main result is a $0.67$-approximation online algorithm for stochastic bipartite matching, breaking this $1 - {1\over e}$ barrier. Furthermore, we show that no online algorithm can produce a $1-\epsilon$ approximation for an arbitrarily small $\epsilon$ for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-) matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.
326 citations
TL;DR: This paper unifies prominent characterizations of layout quality and clustering quality, by showing that energy models of pairwise attraction and repulsion subsume Newman and Girvan's modularity measure.
Abstract: Two natural and widely used representations for the community structure of networks are clusterings, which partition the vertex set into disjoint subsets, and layouts, which assign the vertices to positions in a metric space. This paper unifies prominent characterizations of layout quality and clustering quality, by showing that energy models of pairwise attraction and repulsion subsume Newman and Girvan's modularity measure. Layouts with optimal energy are relaxations of, and are thus consistent with, clusterings with optimal modularity, which is of practical relevance because the two representations are complementary and often used together.
234 citations
TL;DR: In this paper, the density matrix corresponding to the vacuum state of a massless Dirac field in two dimensions is reduced to a region of the space formed by several disjoint intervals.
Abstract: We find the density matrix corresponding to the vacuum state of a massless Dirac field in two dimensions reduced to a region of the space formed by several disjoint intervals. We calculate explicitly its spectral decomposition. The imaginary power of the density matrix is a unitary operator implementing an internal time flow (the modular flow). We show that in the case of more than one interval, this evolution is non-local, producing both advance in the causal structure and 'teleportation' between the disjoint intervals. However, it only mixes the fields on a finite number of trajectories, one for each interval. As an application of these results we compute the entanglement entropy for the massive multi-interval case in the small mass limit.
207 citations
TL;DR: In this paper, a bivariate process Xt = (X 1 t ; X 2 t ), which is observed on a flnite time interval [0;T], at discrete times 0;¢n;2¢n,¢¢¢.
Abstract: We consider a bivariate process Xt = (X 1 t ;X 2 t ), which is observed on a flnite time interval [0;T], at discrete times 0;¢n;2¢n;¢¢¢. Assuming that its two components X 1 and X 2 have jumps on [0;T], we derive tests to decide whether they have at least one jump occurring at the same time (\common jumps") or not (\disjoint jumps"). There are two difierent tests, for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level, as the mesh ¢n goes to 0. We show on some simulations that these tests perform reasonably well even in the flnite sample case, and we also put them in use for some exchange rates data.
195 citations
TL;DR: The novel idea of hypothesized source locations in the algorithm development is introduced to enable the formulation of psuedolinear equations, thereby leading to the establishment of closed-form solution for source location estimates.
Abstract: This paper considers the problem of time difference-of-arrival (TDOA) source localization when the TDOA measurements from multiple disjoint sources are subject to the same sensor position displacements from the available sensor positions. This is a challenging problem and closed-form solution with good localization accuracy has yet to be found. This paper proposes an estimator that can achieve this purpose. The proposed algorithm jointly estimates the unknown source and sensor positions to take the advantage that the TDOAs from different sources have the same sensor position displacements. The joint estimation is a highly nonlinear problem due to the coupling of source and sensor positions in the measurement equations. We introduce the novel idea of hypothesized source locations in the algorithm development to enable the formulation of psuedolinear equations, thereby leading to the establishment of closed-form solution for source location estimates. Besides the advantage of closed-form, the newly developed algorithm is shown analytically, under the condition that the TDOA measurement noise and the sensor position errors are sufficiently small, to reach the CRLB accuracy. For clarity, the localization of two disjoint sources is used in the algorithm development. The developed algorithm is then examined under the special case of a single source and extended to the more general case of more than two unknown sources. The theoretical developments are supported by simulations.
186 citations
TL;DR: A newly developed ''absorbing'' technique is employed, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.
178 citations
TL;DR: In this article, the authors give an overview of recent progress in F-packings, with the main emphasis on F-packing, Hamiltonicity problems and tree embeddings, and describe some of the methods involved.
Abstract: What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.
176 citations
03 Jul 2009
TL;DR: A 2e -competitive algorithm is given for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph.
Abstract: We examine online matching problems with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G (L *** R , E ). We develop an 8-competitive algorithm for the following secretary problem: Initially given R , and the size of L , the algorithm receives the vertices of L sequentially, in a random order. When a vertex l *** L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R , or decide that l will remain unmatched.
In [5], the authors show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l *** L , the weights on all edges incident to l are identical.) We use a very similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Our analysis is easily extended to obtain competitive algorithms for a class of similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups .
Finally, we give a 2e -competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph.
07 Sep 2009
TL;DR: Evidence is given that DisJoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coNP/poly, and it is shown that the related Disj Joint Cycles Packing problem has a kernel of size O(k logk).
Abstract: In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic assumptions. We build upon recent results by Bodlaender et al. [3] and Fortnow and Santhanam [13], that show that NP-complete problems that are or-compositional do not have polynomial kernels, unless NP ⊆ coNP/poly. To this machinery, we add a notion of transformation, and thus obtain that Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coNP/poly. We also show that the related Disjoint Cycles Packing problem has a kernel of size O(k logk).
TL;DR: In this article, the authors examined the class of barotropic fluid models of dark energy, in which the pressure is an explicit function of the density, p=f(rho) and showed that this class is equivalent to the sum of a cosmological constant and a decelerating perfect fluid, or "aether", with w{sub AE}>=}0.
Abstract: We examine the class of barotropic fluid models of dark energy, in which the pressure is an explicit function of the density, p=f({rho}). Through general physical considerations we constrain the asymptotic past and future behaviors and show that this class is equivalent to the sum of a cosmological constant and a decelerating perfect fluid, or 'aether', with w{sub AE}{>=}0. Barotropic models give substantially disjoint predictions from quintessence, except in the limit of {lambda}CDM. They are also interesting in that they simultaneously can ameliorate the coincidence problem and yet 'predict' a value of w{approx_equal}-1.
04 Jan 2009
TL;DR: The main result of this paper is an O(log log n)-approximation algorithm for MISR, which combines existing approaches for solving special cases of the problem, in which the input set of rectangles is restricted to containing specific intersection types, with new insights into the combinatorial structure of sets of intersecting rectangles in the plane.
Abstract: We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axis-parallel rectangles, find a maximum-cardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection graphs of axis-parallel rectangles. Due to its many applications, ranging from map labeling to data mining, MISR has received a significant amount of attention from various research communities. Since the problem is NP-hard, the main focus has been on the design of approximation algorithms. Several groups of researches have independently suggested O(log n)-approximation algorithms for MISR, and this remained the best currently known approximation factor for the problem. The main result of our paper is an O(log log n)-approximation algorithm for MISR. Our algorithm combines existing approaches for solving special cases of the problem, in which the input set of rectangles is restricted to containing specific intersection types, with new insights into the combinatorial structure of sets of intersecting rectangles in the plane.We also consider a generalization of MISR to higher dimensions, where rectangles are replaced by d-dimensional hyper-rectangles. Our results for MISR imply an O((log n)d−2 log log n)-approximation algorithm for this problem, improving upon the best previously known O((log n)d−1)-approximation.
25 Oct 2009
TL;DR: A long code test with one free bit, completeness 1-epsilon and soundness delta is presented, and the following two inapproximability results are proved.
Abstract: For arbitrarily small constants epsilon, delta ≫ 0$, we present a long code test with one free bit, completeness 1-epsilon and soundness delta. Using the test, we prove the following two inapproximability results:1. Assuming the Unique Games Conjecture of Khot, given an n-vertex graph that has two disjoint independent sets of size (1/2-epsilon)n each, it is NP-hard to find an independent set of size delta n.2. Assuming a (new) stronger version of the Unique Games Conjecture, the scheduling problem of minimizing weighted completion time with precedence constraints is inapproximable within factor 2-epsilon.
01 Oct 2009
TL;DR: In this article, it was shown that if the ground set of a represented linear matroid is partitioned into blocks of size @?, then we can determine in randomized time f(k,@?)@?n^O^(^1^) whether there is an independent set that is the union of k blocks.
Abstract: Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomial-time algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial n^O^(^k^) time brute force algorithm for finding a k-element solution, we try to give algorithms with uniformly polynomial (i.e., f(k)@?n^O^(^1^)) running time. The main result is that if the ground set of a represented linear matroid is partitioned into blocks of size @?, then we can determine in randomized time f(k,@?)@?n^O^(^1^) whether there is an independent set that is the union of k blocks. As a consequence, algorithms with similar running time are obtained for other problems such as finding a k-element set in the intersection of @? matroids, or finding k terminals in a network such that each of them can be connected simultaneously to the source by @? disjoint paths.
TL;DR: This work proposes a new, two-phase, method of detecting overlapping communities, which has the potential to convert any disjoint community detection algorithm into an overlappingcommunity detection algorithm.
Abstract: Many algorithms have been designed to discover community structure in networks. Most of these detect disjoint communities, while a few can find communities that overlap. We propose a new, two-phase, method of detecting overlapping communities. In the first phase, a network is transformed to a new one by splitting vertices, using the idea of split betweenness; in the second phase, the transformed network is processed by a disjoint community detection algorithm. This approach has the potential to convert any disjoint community detection algorithm into an overlapping community detection algorithm. Our experiments, using several “disjoint” algorithms, demonstrate that the method works, producing solutions, and execution times, that are often better than those produced by specialized “overlapping” algorithms.
TL;DR: In this article, it was shown that the problem of embedding a rational ellipsoid into another is equivalent to embedding disjoint equal balls into ℂP 2, where k is the ratio of the area of the major axis to that of the minor axis.
Abstract: We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into ℂP 2 . For example, the problem of embedding the ellipsoid E(1, k) into a ball B is equivalent to that of embedding k disjoint equal balls into ℂP 2 , and so can be solved by the work of Gromov, McDuff-Polterovich, and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1, k) for k = 1, 4 and all k ⩾ 9, thus answering a question raised by Hofer.
TL;DR: A constrained principal component analysis, which aims at a simultaneous clustering of objects and a partitioning of variables, is proposed, formulated in a semi-parametric least-squares framework as a quadratic mixed continuous and integer problem.
TL;DR: Menger's theorem is valid for infinite graphs in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph, and then there exist a set of disjoint A-B paths, and a set S of nodes separating A from B, such that S consists of a choice of precisely one vertex from each path as mentioned in this paper.
Abstract: We prove that Menger’s theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set \(\mathcal{P}\) of disjoint A–B paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in \(\mathcal{P}\). This settles an old conjecture of Erdős.
TL;DR: This work presents good combinatorial algorithms for solving k-clique problems that do not require large constants in their runtime, can be readily implemented in any reasonable random access model, and are very space-efficient compared to their algebraic counterparts.
TL;DR: Using the construction, it is shown that every m-dimensional restricted HL-graph and recursive circulant G(2m, 4) with f or less faulty elements have a paired k-DPC for any f and k ges 2 with f + 2k les m.
Abstract: A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k sources and k sinks in which each vertex of G is covered by a path. It is called a paired many-to-many disjoint path cover when each source should be joined to a specific sink, and it is called an unpaired many-to-many disjoint path cover when each source can be joined to an arbitrary sink. In this paper, we discuss about paired and unpaired many-to-many disjoint path covers including their relationships, application to strong Hamiltonicity, and necessary conditions. And then, we give a construction scheme for paired many-to-many disjoint path covers in the graph H0 oplus H1 obtained from connecting two graphs H0 and H1 with |V(H0)| = |V(H1)| by |V(H1)| pairwise nonadjacent edges joining vertices in H0 and vertices in H1, where H0 = G0 oplus G1 and H1 = G2 oplus G3 for some graphs Gj. Using the construction, we show that every m-dimensional restricted HL-graph and recursive circulant G(2m, 4) with f or less faulty elements have a paired k-DPC for any f and k ges 2 with f + 2k les m.
TL;DR: Here it is shown that the use of submodular flows is actually avoidable and even a common generalization of the two rooted k-connection problems reduces to matroid intersection and the approach is based on a new matroid construction extending what Whiteley calls count matroids.
TL;DR: It is shown that given any m, one can construct infinitely many re-variable (n even), m-resilient functions with nonlinearity >2n-1 - 2n-2.
Abstract: In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any m, one can construct infinitely many re-variable (n even), m-resilient functions with nonlinearity > 2n-1 - 2n-2. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.
01 Oct 2009
TL;DR: The presented algorithm for finding maximum weight matchings in bipartite graphs with nonnegative integer weights works in [email protected]?(Wn^@w) time, where @w is the matrix multiplication exponent, and W is the highest edge weight in the graph.
Abstract: In this paper we consider the problem of finding maximum weight matchings in bipartite graphs with nonnegative integer weights. The presented algorithm for this problem works in [email protected]?(Wn^@w) time, where @w is the matrix multiplication exponent, and W is the highest edge weight in the graph. As a consequence of this result we obtain [email protected]?(Wn^@w) time algorithms for computing: minimum weight bipartite vertex cover, single source shortest paths and minimum weight vertex disjoint s-t paths. All of the presented algorithms are randomized and with small probability can return suboptimal solutions.
TL;DR: In this article, it was shown that the symmetric entropy formula describing black holes and black strings in $D=5$ is intimately tied to the geometry of the generalized quadrangle GQ(2, 4) with automorphism group the Weyl group.
Abstract: It is shown that the ${E}_{6(6)}$ symmetric entropy formula describing black holes and black strings in $D=5$ is intimately tied to the geometry of the generalized quadrangle GQ(2, 4) with automorphism group the Weyl group $W({E}_{6})$. The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ(2, 4). Different truncations with 15, 11 and 9 charges are represented by three distinguished subconfigurations of GQ(2, 4), well known to finite geometers; these are the ``doily'' [i.e. GQ(2, 2)] with 15, the ``perp set'' of a point with 11, and the ``grid'' [i.e. GQ(2, 1)] with nine points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a noncommutative labeling for the points of GQ(2, 4). For the 40 different possible truncations with nine charges this labeling yields 120 Mermin squares---objects well known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the ${E}_{7(7)}$ symmetric entropy formula in $D=4$ by observing that the structure of GQ(2, 4) is linked to a particular kind of geometric hyperplane of the split Cayley hexagon of order 2, featuring 27 points located on nine pairwise disjoint lines (a distance-3-spread). We conjecture that the different possibilities of describing the $D=5$ entropy formula using Jordan algebras, qubits and/or qutrits correspond to employing different coordinates for an underlying noncommutative geometric structure based on GQ(2, 4).
TL;DR: A distributed algorithm for constructing the connected domatic partition (CDP) problem is developed using the maximal independent set (MlS)-based proximity heuristics, which depends only on connectivity information and does not rely on geographic or geometric information.
Abstract: Wireless ad hoc and sensor networks (WSNs) often require a connected dominating set (CDS) as the underlying virtual backbone for efficient routing. Nodes in a CDS have extra computation and communication load for their role as dominator, subjecting them to an early exhaustion of their battery. A simple mechanism to address this problem is to switch from one CDS to another fresh CDS, rotating the active CDS through a disjoint set of CDSs. This gives rise to the connected domatic partition (CDP) problem, which essentially involves partitioning the nodes V(G) of a graph G into node disjoint CDSs. We have developed a distributed algorithm for constructing the CDP using our maximal independent set (MlS)-based proximity heuristics, which depends only on connectivity information and does not rely on geographic or geometric information. We show that the size of a CDP that is identified by our algorithm is at least [delta+1/beta(c+1)] - f, where delta is the minimum node degree of G, beta les 2, c les 11 is a constant for a unit disk graph (UDG), and the expected value of f is epsidelta|V|, where epsi Lt 1 is a positive constant, and delta ges 48. Results of varied testing of our algorithm are positive even for a network of a large number of sensor nodes. Our scheme also performs better than other related techniques such as the ID-based scheme.
TL;DR: In this article, the complement of a union of at least three disjoint (round) open balls in the unit sphere is called a Schottky set, and it is shown that every quasisymmetric homeomorphism of such a set is the restriction of a M*obius transformation.
Abstract: We call the complement of a union of at least three disjoint (round)
open balls in the unit sphere ${\Bbb S}^n$ a Schottky set. We prove
that every quasisymmetric homeomorphism of a Schottky set of
spherical measure zero to another Schottky set is the restriction
of a M\"obius transformation on ${\Bbb S}^n$. In the other direction we
show that every Schottky set in ${\Bbb S}^2$ of positive measure admits
nontrivial quasisymmetric maps to other Schottky sets. These results
are applied to establish rigidity statements for convex subsets of
hyperbolic space that have totally geodesic boundaries.
TL;DR: This paper considers the problem of many-to-many disjoint paths in the hypercube Q n with f v faulty vertices and f e faulty edges, and obtains the following result, which is optimal in the worst case.
07 Sep 2009
TL;DR: It is shown that one can count k-edge paths in an n-vertex graph and m-set k-packings on ann-element universe, respectively, in time up to a factor polynomial in n, k, and m; inPolynomial space, the bounds hold if multiplied by 3 k/2 or 5 mk/2, respectively.
Abstract: We show that one can count k-edge paths in an n-vertex graph and m-set k-packings on an n-element universe, respectively, in time \({n \choose k/2}\) and \({n \choose mk/2}\), up to a factor polynomial in n, k, and m; in polynomial space, the bounds hold if multiplied by 3 k/2 or 5 mk/2, respectively. These are implications of a more general result: given two set families on an n-element universe, one can count the disjoint pairs of sets in the Cartesian product of the two families with O(n l) basic operations, where l is the number of members in the two families and their subsets.