Showing papers on "Disjoint sets published in 2013"

The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT

Abstract: We study entanglement Renyi entropies (EREs) of 1+1 dimensional CFTs with classical gravity duals. Using the replica trick the EREs can be related to a partition function of n copies of the CFT glued together in a particular way along the intervals. In the case of two intervals this procedure defines a genus n-1 surface and our goal is to find smooth three dimensional gravitational solutions with this surface living at the boundary. We find two families of handlebody solutions labelled by the replica index n. These particular bulk solutions are distinguished by the fact that they do not spontaneously break the replica symmetries of the boundary surface. We show that the regularized classical action of these solutions is given in terms of a simple numerical prescription. If we assume that they give the dominant contribution to the gravity partition function we can relate this classical action to the EREs at leading order in G_N. We argue that the prescription can be formulated for non-integer n. Upon taking the limit n -> 1 the classical action reproduces the predictions of the Ryu-Takayanagi formula for the entanglement entropy.

Holographic entanglement beyond classical gravity

Abstract: The Renyi entropies and entanglement entropy of 1+1 CFTs with gravity duals can be computed by explicit construction of the bulk spacetimes dual to branched covers of the boundary geometry. At the classical level in the bulk this has recently been shown to reproduce the conjectured Ryu-Takayanagi formula for the holographic entanglement entropy. We study the one-loop bulk corrections to this formula. The functional determinants in the bulk geometries are given by a sum over certain words of generators of the Schottky group of the branched cover. For the case of two disjoint intervals on a line we obtain analytic answers for the one-loop entanglement entropy in an expansion in small cross-ratio. These reproduce and go beyond anticipated universal terms that are not visible classically in the bulk. We also consider the case of a single interval on a circle at finite temperature. At high temperatures we show that the one-loop contributions introduce expected finite size corrections to the entanglement entropy that are not present classically. At low temperatures, the one-loop corrections capture the mixed nature of the density matrix, also not visible classically below the Hawking-Page temperature.

Entanglement negativity in extended systems: a field theoretical approach

Abstract: We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A1∪A2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths l1,l2 in an infinite system, we derive the result ℰ ∼ (c/4)ln(l1l2/(l1 + l2)), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.

Some results on the mutual information of disjoint regions in higher dimensions

Abstract: We consider the mutual Renyi information of disjoint compact spatial regions A and B in the ground state of a d + 1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes RA, B. We show that in general , where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where α = d − 1, we show that is proportional to the capacitance of a thin conducting slab in the shape of A in d + 1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere Sd − 1 or an ellipsoid. For spherical regions in d = 2 and 3 we obtain explicit results for C(n) for all n and hence for the leading term in the mutual information by taking n → 1. We also compute a universal logarithmic correction to the area law for the Renyi entropies of a single spherical region for a scalar field theory with a small mass.

Some Results on Mutual Information of Disjoint Regions in Higher Dimensions

Abstract: We consider the mutual Renyi information I^n(A,B)=S^n_A+S^n_B-S^n_{AUB} of disjoint compact spatial regions A and B in the ground state of a d+1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R_{A,B}. We show that in general I^n(A,B)\sim C^n_AC^n_B(R_AR_B/r^2)^a, where a the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants C^n_{A,B} depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where 2x=d-1, we show that C^2_AR_A^{d-1} is proportional to the capacitance of a thin conducting slab in the shape of A in d+1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S^{d-1} or an ellipsoid. For spherical regions in d=2 and 3 we obtain explicit results for C^n for all n and hence for the leading term in the mutual information by taking n->1. We also compute a universal logarithmic correction to the area law for the Renyi entropies of a single spherical region for a scalar field theory with a small mass.

Restreaming graph partitioning: simple versatile algorithms for advanced balancing

Abstract: Partitioning large graphs is difficult, especially when performed in the limited models of computation afforded to modern large scale computing systems. In this work we introduce restreaming graph partitioning and develop algorithms that scale similarly to streaming partitioning algorithms yet empirically perform as well as fully offline algorithms. In streaming partitioning, graphs are partitioned serially in a single pass. Restreaming partitioning is motivated by scenarios where approximately the same dataset is routinely streamed, making it possible to transform streaming partitioning algorithms into an iterative procedure. This combination of simplicity and powerful performance allows restreaming algorithms to be easily adapted to efficiently tackle more challenging partitioning objectives. In particular, we consider the problem of stratified graph partitioning, where each of many node attribute strata are balanced simultaneously. As such, stratified partitioning is well suited for the study of network effects on social networks, where it is desirable to isolate disjoint dense subgraphs with representative user demographics. To demonstrate, we partition a large social network such that each partition exhibits the same degree distribution in the original graph --- a novel achievement for non-regular graphs. As part of our results, we also observe a fundamental difference in the ease with which social graphs are partitioned when compared to web graphs. Namely, the modular structure of web graphs appears to motivate full offline optimization, whereas the locally dense structure of social graphs precludes significant gains from global manipulations.

ON b-ℊ-OPEN SETS

Abstract: In this paper, a new class of generalized open sets in a grill topological spaces called b-ℊ-open sets, is introduced and studied. This class is contained in the class of semi-pre-ℊ-open sets and contains all semi-ℊ-open sets and all pre-ℊ-open sets.

Disjointness of Moebius from Horocycle Flows

Abstract: We formulate and prove a finite version of Vinogradov’s bilinear sum inequality. We use it together with Ratner’s joinings theorems to prove that the Moebius function is disjoint from discrete horocycle flows on \(\Gamma \setminus S{L}_{2}(\mathbb{R})\), where \(\Gamma \subset S{L}_{2}(\mathbb{R})\) is a lattice.

Verification as Learning Geometric Concepts

Abstract: We formalize the problem of program verification as a learning problem, showing that invariants in program verification can be regarded as geometric concepts in machine learning. Safety properties define bad states: states a program should not reach. Program verification explains why a program’s set of reachable states is disjoint from the set of bad states. In Hoare Logic, these explanations are predicates that form inductive assertions. Using samples for reachable and bad states and by applying well known machine learning algorithms for classification, we are able to generate inductive assertions. By relaxing the search for an exact proof to classifiers, we obtain complexity theoretic improvements. Further, we extend the learning algorithm to obtain a sound procedure that can generate proofs containing invariants that are arbitrary boolean combinations of polynomial inequalities. We have evaluated our approach on a number of challenging benchmarks and the results are promising.

Perfect matchings in 3-uniform hypergraphs with large vertex degree ∗

Abstract: A perfect matching in a $3$-uniform hypergraph on $n=3k$ vertices is a subset of $\frac{n}{3}$ disjoint edges. We prove that if $H$ is a $3$-uniform hypergraph on $n=3k$ vertices such that every vertex belongs to at least ${n-1\choose 2} - {2n/3\choose 2}+1$ edges, then $H$ contains a perfect matching. We give a construction to show that this result is the best possible.

Improved Approximation for 3-Dimensional Matching via Bounded Pathwidth Local Search

Abstract: One of the most natural optimization problems is the k-SET PACKING problem, where given a family of sets of size at most k one should select a maximum size subfamily of pairwise disjoint sets. A special case of 3-SET PACKING is the well known 3-DIMENSIONAL MATCHING problem, which is a maximum hypermatching problem in 3-uniform tripartite hypergraphs. Both problems belong to the Karp's list of 21 NP-complete problems. The best known polynomial time approximation ratio for k-SET PACKING is (k + e)/2 and goes back to the work of Hurkens and Schrijver [SIDMA'89], which gives (1.5+e)-approximation for 3-DIMENSIONAL MATCHING. Those results are obtained by a simple local search algorithm, that uses constant size swaps. The main result of this paper is a new approach to local search for k-SET PACKING where only a special type of swaps is considered, which we call swaps of bounded pathwidth. We show that for a fixed value of k one can search the space of r-size swaps of constant pathwidth in crpoly(|F|) time. Moreover we present an analysis proving that a local search maximum with respect to O(log |F|)-size swaps of constant pathwidth yields a polynomial time (k+1+e)/3-approximation algorithm, improving the best known approximation ratio for k-SET PACKING. In particular we improve the approximation ratio for 3-DIMENSIONAL MATCHING from 3/2+e to 4/3+e.

On R\'enyi entropies of disjoint intervals in conformal field theory

Abstract: We study the R\'enyi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.

Applications of the semi-definite method to the turán density problem for 3-graphs

Abstract: In this paper, we prove several new Turan density results for 3-graphs with independent neighbourhoods. We show: \begin{align*} \pi K_4^-, C_5, F_{3,2}=12/49, \pi K_4^-, F_{3,2}=5/18 \textrm {and} \pi J_4, F_{3,2}=\pi J_5, F_{3,2}=3/8, \end{align*} where Jt is the 3-graph consisting of a single vertex x together with a disjoint set A of size t and all $\binom{|A|}{2}$ 3-edges containing x. We also prove two Turan density results where we forbid certain induced subgraphs: \begin{align*} \pi F_{3,2}, \textrm { induced }K_4^-=3/8 \textrm {and} \pi K_5, 5\textrm {-set spanning exactly 8 edges}=3/4. \end{align*} The latter result is an analogue for K5 of Razborov's result that \begin{align*} \pi K_4, 4\textrm {-set spanning exactly 1 edge}=5/9. \end{align*} We give several new constructions, conjectures and bounds for Turan densities of 3-graphs which should be of interest to researchers in the area. Our main tool is 'Flagmatic', an implementation of Razborov's semi-definite method, which we are making publicly available. In a bid to make the power of Razborov's method more widely accessible, we have tried to make Flagmatic as user-friendly as possible, hoping to remove thereby the major hurdle that needs to be cleared before using the semi-definite method. Finally, we spend some time reflecting on the limitations of our approach, and in particular on which problems we may be unable to solve. Our discussion of the 'complexity barrier' for the semi-definite method may be of general interest.

Kidney exchange in dynamic sparse heterogenous pools

Abstract: The need for kidney exchange arises when a healthy person wishes to donate a kidney but is incompatible with her intended recipient. Two main factors determine compatibility of a donor with a patient: blood-type compatibility and tissue-type compatibility. Two or more incompatible pairs can form a cyclic exchange so that each patient can receive a kidney from a compatible donor. In addition, an exchange can be initiated by a non-directed donor (an altruistic donor who does not designate a particular intended patient), and in this case, a chain of exchanges need not form a closed cycle.Current exchange pools are of moderate size and have a dynamic flavor as pairs enroll over time. Further, they contain many highly sensitized patients, i.e., patients that are very unlikely to be tissue-type compatible with a blood-type compatible donor. One major decision clearinghouses are facing is how often to search for allocations (a set of disjoint exchanges). On one hand, waiting for more pairs to arrive before finding allocations will increase the number of matched pairs, especially with highly sensitized patients, and on the other hand, waiting is costly. This paper studies this intrinsic tradeoff between the waiting time, and the number of pairs matched under a myopic, "current-like", matching algorithm called Chunk Matching (CM) that accumulates a given number of incompatible pairs, or a chunk, before searching for an allocation in the pool that consists of easy and hard to match patients.We perform sensitivity analysis on the chunk size given different types of allocations; first, we first study the performance of CM when it searches for allocations limited to cycles of length 2 and show that if the waiting period between two subsequent match runs is a sub-linear function of the problem size (or the entire time horizon), CM matches approximately the same number of pairs as the online scenario (matching each time a new incompatible pair joins the pool) does. Waiting, however, a linear fraction between every two runs will result in matching linearly more pairs compared to the online scenario. We then analyze CM when cycles of length both 2 and 3 are allowed. We show that for some regimes, sub-linear waiting will result in a linear addition of matches comparing to the online scenario. Finally, we study the efficiency of dynamic matching with chains (chains are initiated by a non-directed donor), and show that in the online scenario, adding one non-directed donor will increase linearly the number of matches that CM will find over the number of matches it will find without a chain.Our results may be of independent interest to the literature on dynamic matching in random graphs. Kidney exchange serves well as an example for which we have distributional information on the underlying graphs, thus we can exploit this information to make analysis and prediction far more accurate than the worst-case analysis can do. We believe our average-case analysis can have implications beyond the kidney exchange and can be applied to other dynamic allocation problems with such distributional information. Further, from a theoretical perspective, our paper initiates a novel direction in matching over time, deviating from the online scenarios.While this paper focuses on kidney exchange, there are many dynamic markets for barter exchange for which our findings apply. There is a growing number of websites that accommodate a marketplace for exchange of goods (often more than 2 goods), e.g. ReadItSwapIt.com and Swap.com. In these markets, the demand for goods, cycle lengths and waiting times play a significant role in efficiency.

Efficient Exploration and Value Function Generalization in Deterministic Systems

Abstract: We consider the problem of reinforcement learning over episodes of a finite-horizon deterministic system and as a solution propose optimistic constraint propagation (OCP), an algorithm designed to synthesize efficient exploration and value function generalization. We establish that when the true value function Q* lies within the hypothesis class Q, OCP selects optimal actions over all but at most dimE[Q] episodes, where dimE denotes the eluder dimension. We establish further efficiency and asymptotic performance guarantees that apply even if Q* does not lie in Q, for the special case where Q is the span of pre-specified indicator functions over disjoint sets.

Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Abstract: Separation is a classical problem asking whether, given two sets belonging to some class, it is possible to separate them by a set from another class. We discuss the separation problem for regular languages. We give a Ptime algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a \(\mathcal{B}\Sigma_1(<)\) sentence can witness that the languages are disjoint. The proof refines an algebraic argument from Almeida and the third author. When separation is possible, we also express a separator by saturating one of the original languages by a suitable congruence. Following the same line, we show that one can as well decide whether two regular languages can be separated by an unambiguous language, albeit with a higher complexity.

Near Sets: An Introduction

Abstract: This article gives a brief overview of near sets. The proposed approach in introducing near sets is to consider a set theory-based form of nearness (proximity) called discrete proximity. There are two basic types of near sets, namely, spatially near sets and descriptively near sets. By endowing a nonempty set with some form of a nearness (proximity) relation, we obtain a structured set called a proximity spaces. Let \({\mathcal{P}(X)}\) denote the set of all subsets of a nonempty set X. One of the oldest forms of nearness relations p (later denoted by δ) was introduced by E. Cech during the mid-1930s, which leads to the discovery of spatially near sets, i.e., those sets that have elements in common. That is, given a proximity space (X, δ), for any subset \({A \in \mathcal{P}(X)}\) , one can discover nonempty nearness collections \({\xi(A) = \{B \in \mathcal{P}(X): A \, \delta \, B\} }\) . Recently, descriptively near sets were introduced as a means of solving classification and pattern recognition problems arising from disjoint sets (i.e, sets with empty spatial intersections) that resemble each other. One discovers descriptively near sets by choosing a set of probe functions Φ that represent features of points in a set and endowing the set of points with a descriptive proximity relation δ Φ and obtaining a descriptively structured set (called descriptive proximity space). Given a descriptive proximity spaces (X, δ Φ), one can discover collections of subsets that resemble each other. This leads to the discovery of descriptive nearness collections \({\xi_{\Phi}(A) = \{B \in \mathcal{P}(X): A \,\delta_{\Phi} \, B\} }\) . That is, if \({B \in \xi_{\Phi}(A)}\) , then A δ Φ B (relative to the chosen features of points in X, A resembles B). The focus of this tutorial is on descriptively near sets.

Improved approximation for 3-dimensional matching via bounded pathwidth local search

Abstract: One of the most natural optimization problems is the k-Set Packing problem, where given a family of sets of size at most k one should select a maximum size subfamily of pairwise disjoint sets. A special case of 3-Set Packing is the well known 3-Dimensional Matching problem. Both problems belong to the Karp`s list of 21 NP-complete problems. The best known polynomial time approximation ratio for k-Set Packing is (k + eps)/2 and goes back to the work of Hurkens and Schrijver [SIDMA`89], which gives (1.5 + eps)-approximation for 3-Dimensional Matching. Those results are obtained by a simple local search algorithm, that uses constant size swaps. The main result of the paper is a new approach to local search for k-Set Packing where only a special type of swaps is considered, which we call swaps of bounded pathwidth. We show that for a fixed value of k one can search the space of r-size swaps of constant pathwidth in c^r poly(|F|) time. Moreover we present an analysis proving that a local search maximum with respect to O(log |F|)-size swaps of constant pathwidth yields a polynomial time (k + 1 + eps)/3-approximation algorithm, improving the best known approximation ratio for k-Set Packing. In particular we improve the approximation ratio for 3-Dimensional Matching from 3/2 + eps to 4/3 + eps.

On the Morse–Sard property and level sets of Sobolev and BV functions

Abstract: We establish Luzin $N$ and Morse-Sard properties for $BV_2$-functions defined on open domains in the plane. Using these results we prove that almost all level sets are finite disjoint unions of Lipschitz arcs whose tangent vectors are of bounded variation. In the case of $W^{2,1}$-functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of $C^1$-arcs whose tangent vectors are absolutely continuous.

Blowup and scattering problems for the nonlinear Schrödinger equations

Abstract: We consider L2-supercritical and H1-subcritical focusing nonlinear[4] Schrodinger equations. We introduce a subset PW of H1(Rd) for d≥1, and investigate behavior of the solutions with initial data in this set. To this end, we divide PW into two disjoint components PW+ and PW−. Then, it turns out that any solution starting from a datum in PW+ behaves asymptotically free, and solution starting from a datum in PW− blows up or grows up, from which we find that the ground state has two unstable directions. Our result is an extension of the one by Duyckaerts, Holmer, and Roudenko to the general powers and dimensions, and our argument mostly follows the idea of Kenig and Merle.

Inverse Domination in Graphs

Abstract: The concept of dominating sets introduced by Ore and Berge, is currently receiving much attention in the literature of graph theory. Several types of domination parameters have been studied by imposing several conditions on dominating sets. Ore observed that the complement of every minimal dominating set of a graph with minimum degree at least one is also a dominating set. This implies that every graph with minimum degree at least one has two disjoint dominating sets. Recently several authors initiated the study of the cardinalities of pairs of disjoint dominating sets in graphs. The inverse domination number is the minimum cardinality of a dominating set whose complement contains a minimum dominating set. Motivated by the inverse domination number, there are studies which deals about two disjoint domination number of a graph.

Interpolation and Sidon Sets for Compact Groups

Abstract: Preface .- Introduction .- Hadamard Sets.- $\epsilon$-Kronecker sets.- Sidon sets: Introduction and decomposition properties.- Characterizations of $I_0$ sets.- Proportional characterizations of Sidon sets.- Decompositions of $I_0$ sets.- Sizes of thin sets.- Sets of zero discrete harmonic density.- Related results.-Open problems.- Appendices (Groups, Probability, Combinatoric results,...).- Bibliography.- Author index.- Subject index.- Index of notation.

Performance analysis of reactive connectivity restoration algorithms for wireless sensor and actor networks

Abstract: Preserving inter-actor connectivity is essential in most wireless sensor and actor network (WSAN) applications as nodes have to collaborate and coordinate their actions against the events reported by the sensors. However, failure of a critical (i.e., cut-vertex) node partitions inter-actor network into disjoint segments and thus hinder network operation. The prime objective of this paper is to analyze the performance of reactive connectivity restoration algorithms for delay-tolerant WSAN applications. First, we provide insights to the state-of-the-art reactive connectivity restoration algorithms. Then, we present a Nearest Non-critical Neighbor (NNN) algorithm; a localized and distributed reactive approach for reconnecting network partitions. In NNN, each actor periodically determine its criticality (i.e., cut-vertex or not) based on 2-hop information and exchange with its neighbors. In case of a critical actor failure, the neighbors detect and trigger a connectivity restoration procedure that involves controlled and coordinated node relocation. NNN prefer to displace non-critical nodes during relocation in order to minimize recovery overhead in terms of distance movement and message coordination. We analyze the performance of reactive schemes through theoretical analysis and simulations.

Independence in Database Relations

Abstract: We investigate the implication problem for independence atoms $X \bot Y$ of disjoint attribute sets X and Y on database schemata. A relation satisfies $X \bot Y$ if for every X-value and every Y-value that occurs in the relation there is some tuple in the relation in which the X-value occurs together with the Y-value. We establish an axiomatization by a finite set of Horn rules, and derive an algorithm for deciding the implication problem in low-degree polynomial time in the input. We show how to construct Armstrong relations which satisfy an arbitrarily given set of independence atoms and violate every independence atom not implied by the given set. Our results establish independence atoms as an efficient subclass of embedded multivalued data dependencies which are not axiomatizable by a finite set of Horn rules, and whose implication problem is undecidable.

From Approximate Factorization to Root Isolation with Application to Cylindrical Algebraic Decomposition

Abstract: We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise disjoint disks each containing one of the distinct roots of $p$, and its multiplicity. The algorithm uses approximate factorization as a subroutine. In addition, we apply the new root isolation algorithm to a recent algorithm for computing the topology of a real planar algebraic curve specified as the zero set of a bivariate integer polynomial and for isolating the real solutions of a bivariate polynomial system. For input polynomials of degree $n$ and bitsize $\tau$, we improve the currently best running time from $\tO(n^{9}\tau+n^{8}\tau^{2})$ (deterministic) to $\tO(n^{6}+n^{5}\tau)$ (randomized) for topology computation and from $\tO(n^{8}+n^{7}\tau)$ (deterministic) to $\tO(n^{6}+n^{5}\tau)$ (randomized) for solving bivariate systems.