Showing papers on "Disjoint sets published in 2014"

A Family of Optimal Locally Recoverable Codes

Abstract: A code over a finite alphabet is called locally recoverable (LRC) if every symbol in the encoding is a function of a small number (at most r ) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to a Reed-Solomon code if the locality parameter r is set to be equal to the code dimension. The size of the code alphabet for most parameters is only slightly greater than the code length. The recovery procedure is performed by polynomial interpolation over r points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data (“hot data”).

Efficient computation of representative sets with applications in parameterized and exact algorithms

Abstract: Let M = (E, I) be a matroid and let S = {S1, ...,St} be a family of subsets of E of size p. A subfamily S ⊆ S is q-representative for S if for every set Y ⊆ E of size at most q, if there is a set X e S disjoint from Y with X ∪ Y e I, then there is a set X e S disjoint from Y with X ∪ Y e I. By the classical result of Bollobas, in a uniform matroid, every family of sets of size p has a q-representative family with at most (p+qp) sets. In his famous "two families theorem" from 1977, Lovasz proved that the same bound also holds for any matroid representable over a field F. As observed by Marx, Lovasz's proof is constructive. In this paper we show how Lovasz's proof can be turned into an algorithm constructing a q-representative family of size at most (p+qp) in time bounded by a polynomial in (p+qp), t, and the time required for field operations.We demonstrate how the efficient construction of representative families can be a powerful tool for designing single-exponential parameterized (2O(k)) and exact exponential time (2O(k)) algorithms. The applications of our approach include the following.• In the Long Directed Cycle problem the input is a directed n-vertex graph G and the positive integer k. The task is to find a directed cycle of length at least k in G, if such a cycle exists. As a consequence of our 8k+onO(1) time algorithm, we have that a directed cycle of length at least log n, if such cycle exists, can be found in polynomial time. As it was shown by Bjorklund, Husfeldt, and Khanna [ICALP 2004], under an appropriate complexity assumption, it is impossible to improve this guarantee by more than a constant factor. Thus our algorithm not only improves over the best previous log n/log log n bound of Gabow and Nie [SODA 2004] but also closes the gap between known lower and upper bounds for this problem.• In the Minimum Equivalent Graph (MEG) problem we are seeking a spanning subdigraph D' of a given n-vertex digraph D with as few arcs as possible in which the reachability relation is the same as in the original digraph D. The existence of a single-exponential cn-time algorithm for some constant c > 1 for MEG was open since the work of Moyles and Thompson [JACM 1969].• To demonstrate the diversity of applications of the approach, we provide an alternative proof of the results recently obtained by Bodlaender, Cygan, Kratsch and Nederlof for algorithms on graphs of bounded treewidth, who showed that many "connectivity" problems such as Hamiltonian Cycle or Steiner Tree can be solved in time 2O(t)n on n-vertex graphs of treewidth at most t. We believe that expressing graph problems in "matroid language" shed light on what makes it possible to solve connectivity problems single-exponential time parameterized by treewidth.For the special case of uniform matroids on n elements, we give a faster algorithm computing a representative family in time O((p+q/q)q · 2o(p+q) · t · log n). We use this algorithm to provide the fastest known deterministic parameterized algorithms for k-Path, k-Tree, and more generally, for k-Subgraph Isomorphism, where the k-vertex pattern graph is of constant treewidth. For example, our k-Path algorithm runs in time O(2.851kn log2n log W) on weighted graphs with maximum edge weight W.

The second eigenvalue of the fractional $p-$Laplacian

Abstract: We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem \[ \inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}. \] We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.

Scaling Private Set Intersection to Billion-Element Sets

Abstract: We examine the feasibility of private set intersection (PSI) over massive datasets. PSI, which allows two parties to find the intersection of their sets without revealing them to each other, has numerous applications including to privacy-preserving data mining, location-based services and genomic computations. Unfortunately, the most efficient constructions only scale to sets containing a few thousand elements—even in the semi-honest model and over a LAN.

Sparse Recovery of Streaming Signals Using $\ell_1$ -Homotopy

Abstract: Most of the existing sparse-recovery methods assume a static system: the signal is a finite-length vector for which a fixed set of measurements and sparse representation are available and an l1 problem is solved for the reconstruction. However, the same representation and reconstruction framework is not readily applicable in a streaming system: the signal changes over time, and it is measured and reconstructed sequentially over small intervals. This is particularly desired when dividing signals into disjoint blocks and processing each block separately is infeasible or inefficient. In this paper, we discuss two streaming systems and a new homotopy algorithm for quickly solving the associated l1 problems: 1) recovery of smooth, time-varying signals for which, instead of using block transforms, we use lapped orthogonal transforms for sparse representation and 2) recovery of sparse, time-varying signals that follows a linear dynamic model. For both systems, we iteratively process measurements over a sliding interval and solve a weighted l1-norm minimization problem for estimating sparse coefficients. Since we estimate overlapping portions of the signal while adding and removing measurements, instead of solving a new l1 program as the system changes, we use available signal estimates as starting point in a homotopy formulation and update the solution in a few simple steps. We demonstrate with numerical experiments that our proposed streaming recovery framework provides better reconstruction compared to the methods that represent and reconstruct signals as independent, disjoint blocks, and that our proposed homotopy algorithm updates the solution faster than the current state-of-the-art solvers.

Structure-constrained low-rank representation.

Abstract: Benefiting from its effectiveness in subspace segmentation, low-rank representation (LRR) and its variations have many applications in computer vision and pattern recognition, such as motion segmentation, image segmentation, saliency detection, and semisupervised learning. It is known that the standard LRR can only work well under the assumption that all the subspaces are independent. However, this assumption cannot be guaranteed in real-world problems. This paper addresses this problem and provides an extension of LRR, named structure-constrained LRR (SC-LRR), to analyze the structure of multiple disjoint subspaces, which is more general for real vision data. We prove that the relationship of multiple linear disjoint subspaces can be exactly revealed by SC-LRR, with a predefined weight matrix. As a nontrivial byproduct, we also illustrate that SC-LRR can be applied for semisupervised learning. The experimental results on different types of vision problems demonstrate the effectiveness of our proposed method.

Binary cyclic codes that are locally repairable

Abstract: Codes for storage systems aim to minimize the repair locality, which is the number of disks (or nodes) that participate in the repair of a single failed disk. Simultaneously, the code must sustain a high rate, operate on a small finite field to be practically significant and be tolerant to a large number of erasures. To this end, we construct new families of binary linear codes that have an optimal dimension (rate) for a given minimum distance and locality. Specifically, we construct cyclic codes that are locally repairable for locality 2 and distances 2, 6 and 10. In doing so, we discover new upper bounds on the code dimension, and prove the optimality of enabling local repair by provisioning disjoint groups of disks. Finally, we extend our construction to build codes that have multiple repair sets for each disk.

On Rényi entropies of disjoint intervals in conformal field theory

Abstract: We study the Renyi entropies of N disjoint intervals in the conformal field theories describing 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 state computations agree with the conformal field theory result once the finite size corrections have been taken into account.

Clustering partially observed graphs via convex optimization

Abstract: This paper considers the problem of clustering a partially observed unweighted graph--i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"--i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) low-rank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition/Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors.

Bounds on locally recoverable codes with multiple recovering sets

Abstract: A locally recoverable code (LRC code) is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. Bounds on the rate and distance of such codes have been extensively studied in the literature. In this paper we derive upper bounds on the rate and distance of codes in which every symbol has t ≥ 1 disjoint recovering sets.

A family of optimal locally recoverable codes

Abstract: A code over a finite alphabet is called locally recoverable (LRC) if every symbol in the encoding is a function of a small number (at most r ) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to a Reed-Solomon code if the locality parameter r is set to be equal to the code dimension. The size of the code alphabet for most parameters is only slightly greater than the code length. The recovery procedure is performed by polynomial interpolation over r points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data (“hot data”).

A Dirichlet principle for non reversible Markov chains and some recurrence theorems

Abstract: We extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an application we prove a some recurrence theorems. In particular, we show the recurrence of two-dimensional cycle random walks under a second moment condition on the winding numbers.

Locality and availability in distributed storage

Abstract: This paper studies the problem of information symbol availability in codes: we refer to a systematic code as code with $(r, t)$ -availability if every information (systematic) symbol can be reconstructed from $t$ disjoint groups of other code symbols, each of the sizes at most $r$ . This paper shows that it is possible to construct codes that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant. It further shows that this is possible with the minimum Hamming distance arbitrarily close to the Singleton bound. This paper also presents a bound demonstrating a tradeoff between rate, minimum Hamming distance, and availability parameters. Our codes match the aforementioned bound, and their constructions rely on certain combinatorial structures. Resolvable designs provide one way to realize these required combinatorial structures. The two constructions presented in this paper require field sizes, which are linear and exponential in the code length, respectively. From a practical standpoint, our codes are relevant for distributed storage applications involving hot data, i.e., the information, which is frequently accessed by multiple processes in parallel.

Conformal Blocks and Negativity at Large Central Charge

Abstract: We consider entanglement negativity for two disjoint intervals in 1+1 dimensional CFT in the limit of large central charge. As the two intervals get close, the leading behavior of negativity is given by the logarithm of the conformal block where a set of approximately null descendants appears in the intermediate channel. We compute this quantity numerically and compare with existing analytic methods which provide perturbative expansion in powers of the cross-ratio.

Tverberg plus constraints

Abstract: Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this we introduce a proof technique that combines a concept of “Tverberg unavoidable subcomplexes” with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Živaljevic and Vrecica (1992). We also get a new strengthened version of the generalized van Kampen–Flores theorem by Sarkaria (1991) and Volovikov (1996), an affine version of their “j-wise disjoint” Tverberg theorem, and a topological version of Soberon’s (2013) result on Tverberg points with equal barycentric coordinates.

Guaranteed clustering and biclustering via semidefinite programming

Abstract: Identifying clusters of similar objects in data plays a significant role in a wide range of applications As a model problem for clustering, we consider the densest $$k$$ k -disjoint-clique problem, whose goal is to identify the collection of $$k$$ k disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques In this paper, we establish conditions ensuring exact recovery of the densest $$k$$ k cliques of a given graph from the optimal solution of a particular semidefinite program In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of $$k$$ k large, distinct clusters and a smaller number of outliers This approach also yields a semidefinite relaxation with similar recovery guarantees for the biclustering problem Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels This problem may be posed as that of partitioning the nodes of a weighted bipartite complete graph such that the sum of the densities of the resulting bipartite complete subgraphs is maximized As in our analysis of the densest $$k$$ k -disjoint-clique problem, we show that the correct partition of the objects and features can be recovered from the optimal solution of a semidefinite program in the case that the given data consists of several disjoint sets of objects exhibiting similar features Empirical evidence from numerical experiments supporting these theoretical guarantees is also provided

Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two

Abstract: In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.$$ where p > 1, $${\Omega \subset \mathbb{R}^2}$$ is a bounded domain, $${\Omega_i^{+}}$$ and $${\Omega_j^{-}}$$ are mutually disjoint subdomains of Ω and $${\chi_{\Omega_i^{+}} ({\rm resp}.\; \chi_{\Omega_j^{-}})}$$ are characteristic functions of $${\Omega_i^{+}({\rm resp}. \;\Omega_j^{-}})$$ , q is a harmonic function. We show that if Ω is a simply-connected smooth domain, then for any given C 1-stable critical point of Kirchhoff–Routh function $${\mathcal{W}\;(x_1^{+},\ldots, x_m^{+}, x_1^{-}, \ldots, x_n^{-})}$$ with $${\kappa^{+}_i > 0\,(i = 1,\ldots, m)}$$ and $${\kappa^{-}_j > 0\,(j = 1,\ldots,n)}$$ , there is a stationary classical solution approximating stationary m + n points vortex solution of incompressible Euler equations with total vorticity $${\sum_{i=1}^m \kappa^{+}_i -\sum_{j=1}^n \kappa_j^{-}}$$ . The case that n = 0 can be dealt with in the same way as well by taking each $${\Omega_j^{-}}$$ as an empty set and set $${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$$ .

Big-M Based MIQP Method for Economic Dispatch With Disjoint Prohibited Zones

Abstract: This paper presents a novel big-M based mixed integer quadratic programming (MIQP) method to solve economic load dispatch problem with disjoint prohibited zones. By adding artificial 0-1 binary variables for each prohibited operating zone of generators and employing a binary coding scheme, disjoint feasible regions are represented by complementary linear constraints. Compared to existing MIQP method, the proposed method can achieve global optimal solution with much reduced problem complexity.

Fold maps with singular value sets of concentric spheres

Abstract: In this paper, we study fold maps from C∞ closed manifolds into Euclidean spaces whose singular value sets are disjoint unions of spheres embedded concentrically. We mainly study homology and homotopy groups of manifolds admitting such maps.

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Abstract: We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary (M,g) from a restriction ΛS,R of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here S and R are open sets in ∂M and the restriction ΛS,R corresponds to the case where the Dirichlet data is supported on R+×S and the Neumann data is measured on R+×R. In the novel case where S‾∩R‾=∅, we show that ΛS,R determines the manifold (M,g) uniquely, assuming that the wave equation is exactly controllable from the set of sources S. Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is, λj≤C‖∂νϕj‖L2(S)2,j=1,2,…, where λj are the Dirichlet eigenvalues and where (ϕj)j=1∞ is an orthonormal basis of the corresponding eigenfunctions.

Compatibility of planck and BICEP2 results in light of inflation

Abstract: We investigate the implications for inflation of the detection of B-modes polarization in the cosmic microwave background by BICEP2. We show that the hypothesis of the primordial origin of the measurement is favored only by the first four band powers, while the others would prefer unreasonably large values of the tensor-to-scalar ratio. Using only those four band powers, we carry out a complete analysis in the cosmological and inflationary slow-roll parameter space using the BICEP2 polarization measurements alone and extract the Bayesian evidences and complexities for all the Encyclopaedia Inflationaris models. This allows us to determine the most probable and simplest BICEP2 inflationary scenarios. Although this list contains the simplest monomial potentials, it also includes many other scenarios, suggesting that focusing model building efforts on large field models only is unjustified at this stage. We demonstrate that the sets of inflationary models preferred by Planck alone and BICEP2 alone are almost disjoint, indicating a clear tension between the two data sets. We address this tension with a Bayesian measure of compatibility between BICEP2 and Planck. We find that for models favored by Planck the two data sets tend to be incompatible, whereas there is moderate evidence of compatibility for the BICEP2 preferred models. As a result, it would be premature to draw any conclusion on the best Planck models, such as Starobinsky and/or Kahler moduli inflation. For the subset of scenarios not exhibiting data sets incompatibility, we update the evidences and complexities using both data sets together.

Proof of the 1-factorization and Hamilton decomposition conjectures IV: exceptional systems for the two cliques case

Abstract: In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we prove results on the decomposition of sparse graphs into path systems. These are used in the proof of (i) and (ii) in the case when $G$ is close to the union of two disjoint cliques.

Monge problem in metric measure spaces with riemannian curvature-dimension condition

Abstract: We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space ( X , d , m ) enjoying the Riemannian curvature-dimension condition RCD ∗ ( K , N ) , with N ∞ . For the first marginal measure, we assume that μ 0 ≪ m . As a corollary, we obtain that the Monge problem and its relaxed version, the Monge–Kantorovich problem, attain the same minimal value. Moreover we prove a structure theorem for d -cyclically monotone sets: neglecting a set of zero m -measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.

Partitioning into expanders

Abstract: Let G = (V, E) be an undirected graph, λk be the kth smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that λk > 0 if and only if G has at most k -- 1 connected components. We prove a robust version of this fact. If λk > 0, then for some 1 ≤ e ≤ k -- 1, V can be partitioned into e sets P1, ..., Pe such that each Pi is a low-conductance set in G and induces a high conductance induced subgraph. In particular, [EQUATION] and [EQUATION]-expander.We make our results algorithmic by designing a simple polynomial time spectral algorithm to find such partitioning of G with a quadratic loss in the inside conductance of Pi's. Unlike the recent results on higher order Cheeger's inequality [6, 9], our results does not use higher order eigenfunctions of G. If there is a sufficiently large gap between λk and λk+1, more precisely if [EQUATION] then our algorithm finds a k partitioning of V into sets P1, ..., Pk such that the induced subgraph G[Pi] has a singnificantly larger conductance than the conductance of Pi in G. Such a partitioning may represent the best k clusterings of G. Our algorithm is a simple local search that only uses the Spectral Partitioning algorithm as a subroutine. We expect to see further applications of this simple algorithm in clustering applications.Let ρ(k) = [EQUATION] be the order k conductance constant of G, in words, ρ(k) is the smallest value of the maximum conductance of any k disjoint subsets of V. Our main technical lemma shows that if (1+e)ρ(k)

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Abstract: Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) self-intersection points of maps from finite simplicial complexes (compact polyhedra) into Rd, and study conditions under which such multiple points can be eliminated. The most classical case is that of embeddings (i.e., maps without double points) of a k-dimensional complex K into R2k. For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen obstruction). For k ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f: K → R2k, which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f. One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign. We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y ∈ Rd an r-fold Tverberg point of a map f: Kk → Rd if y lies in the intersection f(σ1)∩…∩ f(σr) of the images of r pairwise disjoint simplices of K. The analogue of (non-)embeddability that we study is the problem Tverberg r k→d: Given a k-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f:Kk → Rd have an r-fold Tverberg point? Here, we show that for fixed r, k and d of the form d = rm and k = (r−1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings). Our main tool is an r-fold analogue of the Whitney trick: Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y− of opposite intersection sign, we can eliminate y+ and y− by a local isotopy of f. In a subsequent paper, we plan to develop this further and present a generalization of the classical Haefliger--Weber Theorem (which yields a necessary and sufficient condition for embeddability of k-complexes into R, for a wider range of dimensions) to intersection points of higher multiplicity.

Convex optimization for the planted k-disjoint-clique problem

Abstract: We consider the k -disjoint-clique problem. The input is an undirected graph G in which the nodes represent data items, and edges indicate a similarity between the corresponding items. The problem is to find within the graph k disjoint cliques that cover the maximum number of nodes of G. This problem may be understood as a general way to pose the classical ‘clustering’ problem. In clustering, one is given data items and a distance function, and one wishes to partition the data into disjoint clusters of data items, such that the items in each cluster are close to each other. Our formulation additionally allows ‘noise’ nodes to be present in the input data that are not part of any of the cliques. The k -disjoint-clique problem is NP-hard, but we show that a convex relaxation can solve it in polynomial time for input instances constructed in a certain way. The input instances for which our algorithm finds the optimal solution consist of k disjoint large cliques (called ‘planted cliques’) that are then obscured by noise edges inserted either at random or by an adversary, as well as additional nodes not belonging to any of the k planted cliques.

Almost Disjoint Families and Topology

Abstract: An infinite family \({\fancyscript{A}}\subset {\fancyscript{P}}(\omega )\) is almost disjoint (AD) if the intersection of any two distinct elements of \({\fancyscript{A}}\) is finite. We survey recent results on almost disjoint families, concentrating on their applications to topology.