53 research outputs found

    From coordinate subspaces over finite fields to ideal multipartite uniform clutters

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    Take a prime power qq, an integer n≥2n\geq 2, and a coordinate subspace S⊆GF(q)nS\subseteq GF(q)^n over the Galois field GF(q)GF(q). One can associate with SS an nn-partite nn-uniform clutter C\mathcal{C}, where every part has size qq and there is a bijection between the vectors in SS and the members of C\mathcal{C}. In this paper, we determine when the clutter C\mathcal{C} is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether qq is 2,42,4, a higher power of 22, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of C\mathcal{C} depends solely on the underlying matroid of SS. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and τ=2\tau=2 Conjectures for this class of clutters.Comment: 32 pages, 6 figure

    Dyadic linear programming and extensions

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    A rational number is dyadic if it has a finite binary representation p/2kp/2^k, where pp is an integer and kk is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case

    The Edmonds-Giles Conjecture and its Relaxations

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    Given a directed graph, a directed cut is a cut with all arcs oriented in the same direction, and a directed join is a set of arcs which intersects every directed cut at least once. Edmonds and Giles conjectured for all weighted directed graphs, the minimum weight of a directed cut is equal to the maximum size of a packing of directed joins. Unfortunately, the conjecture is false; a counterexample was first given by Schrijver. However its ”dual” statement, that the minimum weight of a dijoin is equal to the maximum number of dicuts in a packing, was shown to be true by Luchessi and Younger. Various relaxations of the conjecture have been considered; Woodall’s conjecture remains open, which asks the same question for unweighted directed graphs, and Edmond- Giles conjecture was shown to be true in the special case of source-sink connected directed graphs. Following these inquries, this thesis explores different relaxations of the Edmond- Giles conjecture

    On packing dijoins in digraphs and weighted digraphs

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    In this paper, we make some progress in addressing Woodall's Conjecture, and the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and weighted digraphs. Let D=(V,A)D=(V,A) be a digraph, and let w∈Z≥0Aw\in \mathbb{Z}^A_{\geq 0}. Suppose every dicut has weight at least τ\tau, for some integer τ≥2\tau\geq 2. Let ρ(τ,D,w):=1τ∑v∈Vmv\rho(\tau,D,w):=\frac{1}{\tau}\sum_{v\in V} m_v, where each mvm_v is the integer in {0,1,…,τ−1}\{0,1,\ldots,\tau-1\} equal to w(δ+(v))−w(δ−(v))w(\delta^+(v))-w(\delta^-(v)) mod τ\tau. In this paper, we prove the following results, amongst others: (1) If w=1w={\bf 1}, then AA can be partitioned into a dijoin and a (τ−1)(\tau-1)-dijoin. (2) If ρ(τ,D,w)∈{0,1}\rho(\tau,D,w)\in \{0,1\}, then there is an equitable ww-weighted packing of dijoins of size τ\tau. (3) If ρ(τ,D,w)=2\rho(\tau,D,w)= 2, then there is a ww-weighted packing of dijoins of size τ\tau. (4) If w=1w={\bf 1}, τ=3\tau=3, and ρ(τ,D,w)=3\rho(\tau,D,w)=3, then AA can be partitioned into three dijoins. Each result is best possible: (1) and (4) do not hold for general ww, (2) does not hold for ρ(τ,D,w)=2\rho(\tau,D,w)=2 even if w=1w={\bf 1}, and (3) does not hold for ρ(τ,D,w)=3\rho(\tau,D,w)=3. The results are rendered possible by a \emph{Decompose, Lift, and Reduce procedure}, which turns (D,w)(D,w) into a set of \emph{sink-regular weighted (τ,τ+1)(\tau,\tau+1)-bipartite digraphs}, each of which is a weighted digraph where every vertex is a sink of weighted degree τ\tau or a source of weighted degree τ,τ+1\tau,\tau+1, and every dicut has weight at least τ\tau. Our results give rise to a number of approaches for resolving Woodall's Conjecture, fixing the refuted Edmonds-Giles Conjecture, and the τ=2\tau=2 Conjecture for the clutter of minimal dijoins. They also show an intriguing connection to Barnette's Conjecture.Comment: 71 page

    Cooperative Radio Communications for Green Smart Environments

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    The demand for mobile connectivity is continuously increasing, and by 2020 Mobile and Wireless Communications will serve not only very dense populations of mobile phones and nomadic computers, but also the expected multiplicity of devices and sensors located in machines, vehicles, health systems and city infrastructures. Future Mobile Networks are then faced with many new scenarios and use cases, which will load the networks with different data traffic patterns, in new or shared spectrum bands, creating new specific requirements. This book addresses both the techniques to model, analyse and optimise the radio links and transmission systems in such scenarios, together with the most advanced radio access, resource management and mobile networking technologies. This text summarises the work performed by more than 500 researchers from more than 120 institutions in Europe, America and Asia, from both academia and industries, within the framework of the COST IC1004 Action on "Cooperative Radio Communications for Green and Smart Environments". The book will have appeal to graduates and researchers in the Radio Communications area, and also to engineers working in the Wireless industry. Topics discussed in this book include: • Radio waves propagation phenomena in diverse urban, indoor, vehicular and body environments• Measurements, characterization, and modelling of radio channels beyond 4G networks• Key issues in Vehicle (V2X) communication• Wireless Body Area Networks, including specific Radio Channel Models for WBANs• Energy efficiency and resource management enhancements in Radio Access Networks• Definitions and models for the virtualised and cloud RAN architectures• Advances on feasible indoor localization and tracking techniques• Recent findings and innovations in antenna systems for communications• Physical Layer Network Coding for next generation wireless systems• Methods and techniques for MIMO Over the Air (OTA) testin

    Clean clutters and dyadic fractional packings

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    A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary cas

    Modelling, Simulation and Data Analysis in Acoustical Problems

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    Modelling and simulation in acoustics is currently gaining importance. In fact, with the development and improvement of innovative computational techniques and with the growing need for predictive models, an impressive boost has been observed in several research and application areas, such as noise control, indoor acoustics, and industrial applications. This led us to the proposal of a special issue about “Modelling, Simulation and Data Analysis in Acoustical Problems”, as we believe in the importance of these topics in modern acoustics’ studies. In total, 81 papers were submitted and 33 of them were published, with an acceptance rate of 37.5%. According to the number of papers submitted, it can be affirmed that this is a trending topic in the scientific and academic community and this special issue will try to provide a future reference for the research that will be developed in coming years

    Sensor Signal and Information Processing II

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    In the current age of information explosion, newly invented technological sensors and software are now tightly integrated with our everyday lives. Many sensor processing algorithms have incorporated some forms of computational intelligence as part of their core framework in problem solving. These algorithms have the capacity to generalize and discover knowledge for themselves and learn new information whenever unseen data are captured. The primary aim of sensor processing is to develop techniques to interpret, understand, and act on information contained in the data. The interest of this book is in developing intelligent signal processing in order to pave the way for smart sensors. This involves mathematical advancement of nonlinear signal processing theory and its applications that extend far beyond traditional techniques. It bridges the boundary between theory and application, developing novel theoretically inspired methodologies targeting both longstanding and emergent signal processing applications. The topic ranges from phishing detection to integration of terrestrial laser scanning, and from fault diagnosis to bio-inspiring filtering. The book will appeal to established practitioners, along with researchers and students in the emerging field of smart sensors processing
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