Restricted rr-Stirling Numbers and their Combinatorial Applications

We study set partitions with $r$ distinguished elements and block sizes found in an arbitrary index set $S$. The enumeration of these $(S,r)$-partitions leads to the introduction of $(S,r)$-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the $r$-Stirling numbers. We also introduce the associated $(S,r)$-Bell and $(S,r)$-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For $S$ with some extra structure, we show that the inverse of the $(S,r)$-Stirling matrix encodes the M\"obius functions of two families of posets. Through several examples, we demonstrate that for some $S$ the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the $(S,r)$-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related $(S,r)$ generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences

New Bounds on Augmenting Steps of Block-Structured Integer Programs

Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4-block n-fold integer programs, which are the most general class considered so far. A 4-block n-fold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the so-called Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the ??- or ?_?-norm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4-block n-fold IP has Graver elements of ?_?-norm at most ?_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and ?_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to ?_FPT(1), perhaps at least in some restricted cases. We prove that the ?_?-norm of the Graver elements of 4-block n-fold IP is upper bounded by ?_FPT(n^{s_D}), improving significantly over the previous bound ?_FPT(n^{2^{s_D}}). We also provide a matching lower bound of ?(n^{s_D}) which even holds for arbitrary non-zero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4-block n-fold in which C is a zero matrix, called 3-block n-fold IP. We show that while the ?_?-norm of its Graver elements is ?(n^{s_D}), there exists a different decomposition into lattice elements whose ?_?-norm is bounded by ?_FPT(1), which allows us to provide improved upper bounds on the ?_?-norm of Graver elements for 3-block n-fold IP. The key difference between the respective decompositions is that a Graver basis guarantees a sign-compatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3-block n-fold IP and 4-block IP, and our lower bounds strongly hint at parameterized hardness of 4-block and even 3-block n-fold IP. Furthermore, we show that 3-block n-fold IP is without loss of generality in the sense that 4-block n-fold IP can be solved in FPT oracle time by taking an algorithm for 3-block n-fold IP as an oracle

The scheduling of sparse matrix-vector multiplication on a massively parallel dap computer

An efficient data structure is presented which supports general unstructured sparse matrix-vector multiplications on a Distributed Array of Processors (DAP). This approach seeks to reduce the inter-processor data movements and organises the operations in batches of massively parallel steps by a heuristic scheduling procedure performed on the host computer. The resulting data structure is of particular relevance to iterative schemes for solving linear systems. Performance results for matrices taken from well known Linear Programming (LP) test problems are presented and analysed

Bipartite Mixed Membership Distribution-Free Model. A novel model for community detection in overlapping bipartite weighted networks

Modeling and estimating mixed memberships for un-directed un-weighted networks in which nodes can belong to multiple communities has been well studied in recent years. However, for a more general case, the bipartite weighted networks in which nodes can belong to multiple communities, row nodes can be different from column nodes, and all elements of adjacency matrices can be any finite real values, to our knowledge, there is no model for such bipartite weighted networks. To close this gap, this paper introduces a novel model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. As a special case, bipartite signed networks with mixed memberships can also be generated from BiMMDF. Our model enjoys its advantage by allowing all elements of an adjacency matrix to be generated from any distribution as long as the expectation adjacency matrix has a block structure related to node memberships under BiMMDF. The proposed model can be viewed as an extension of many previous models, including the popular mixed membership stochastic blcokmodels. An efficient algorithm with a theoretical guarantee of consistent estimation is applied to fit BiMMDF. In particular, for a standard bipartite weighted network with two row (and column) communities, to make the algorithm's error rates small with high probability, separation conditions are obtained when adjacency matrices are generated from different distributions under BiMMDF. The behavior differences of different distributions on separation conditions are verified by extensive synthetic bipartite weighted networks generated under BiMMDF. Experiments on real-world directed weighted networks illustrate the advantage of the algorithm in studying highly mixed nodes and asymmetry between row and column communities.Comment: 33 pages, 12 figures, 4 table

Pade-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

A standard approach to reduced-order modeling of higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are characterized as Pade-type or even true Pade approximants of the system's transfer function, in general, these models do not preserve the form of the original higher-order system. In this paper, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. While the resulting reduced-order models are no longer optimal in the Pade sense, we show that they still satisfy a Pade-type approximation property. We also introduce the notion of Hermitian higher-order linear dynamical systems, and we establish an enhanced Pade-type approximation property in the Hermitian case

A robust spectral method for finding lumpings and meta stable states of non-reversible Markov chains

A spectral method for identifying lumping in large Markov chains is presented. Identification of meta stable states is treated as a special case. The method is based on spectral analysis of a self-adjoint matrix that is a function of the original transition matrix. It is demonstrated that the technique is more robust than existing methods when applied to noisy non-reversible Markov chains.Comment: 10 pages, 7 figure