2,371 research outputs found
Elections with Three Candidates Four Candidates and Beyond: Counting Ties in the Borda Count with Permutahedra and Ehrhart Quasi-Polynomials
In voting theory, the Borda countâs tendency to produce a tie in an election varies as a function of n, the number of voters, and m, the number of candidates. To better understand this tendency, we embed all possible rankings of candidates in a hyperplane sitting in m-dimensional space, to form an (m - 1)-dimensional polytope: the m-permutahedron. The number of possible ties may then be determined computationally using a special class of polynomials with modular coefficients. However, due to the growing complexity of the system, this method has not yet been extended past the case of m = 3. We examine the properties of certain voting situations for m â„ 4 to better understand an electionâs tendency to produce a Borda tie between all candidates
On joint detection and decoding of linear block codes on Gaussian vector channels
Optimal receivers recovering signals transmitted across noisy communication channels employ a maximum-likelihood (ML) criterion to minimize the probability of error. The problem of finding the most likely transmitted symbol is often equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In systems that employ error-correcting coding for data protection, the symbol space forms a sparse lattice, where the sparsity structure is determined by the code. In such systems, ML data recovery may be geometrically interpreted as a search for the closest point in the sparse lattice. In this paper, motivated by the idea of the "sphere decoding" algorithm of Fincke and Pohst, we propose an algorithm that finds the closest point in the sparse lattice to the given vector. This given vector is not arbitrary, but rather is an unknown sparse lattice point that has been perturbed by an additive noise vector whose statistical properties are known. The complexity of the proposed algorithm is thus a random variable. We study its expected value, averaged over the noise and over the lattice. For binary linear block codes, we find the expected complexity in closed form. Simulation results indicate significant performance gains over systems employing separate detection and decoding, yet are obtained at a complexity that is practically feasible over a wide range of system parameters
Exploring Communities in Large Profiled Graphs
Given a graph and a vertex , the community search (CS) problem
aims to efficiently find a subgraph of whose vertices are closely related
to . Communities are prevalent in social and biological networks, and can be
used in product advertisement and social event recommendation. In this paper,
we study profiled community search (PCS), where CS is performed on a profiled
graph. This is a graph in which each vertex has labels arranged in a
hierarchical manner. Extensive experiments show that PCS can identify
communities with themes that are common to their vertices, and is more
effective than existing CS approaches. As a naive solution for PCS is highly
expensive, we have also developed a tree index, which facilitate efficient and
online solutions for PCS
The matrix model for hypergeometric Hurwitz numbers
We present the multi-matrix models that are the generating functions for
branched covers of the complex projective line ramified over fixed points
, , (generalized Grotendieck's dessins d'enfants) of fixed
genus, degree, and the ramification profiles at two points, and . We
take a sum over all possible ramifications at other points with the fixed
length of the profile at and with the fixed total length of profiles at
the remaining points. All these models belong to a class of
hypergeometric Hurwitz models thus being tau functions of the
Kadomtsev--Petviashvili (KP) hierarchy. In the case described above, we can
present the obtained model as a chain of matrices with a (nonstandard)
nearest-neighbor interaction of the type \tr M_iM_{i+1}^{-1}. We describe the
technique for evaluating spectral curves of such models, which opens the
possibility of applying the topological recursion for developing
-expansions of these model. These spectral curves turn out to be of an
algebraic type.Comment: 12 pages, 2 figures in LaTeX, contribution to the volume of TMPh
celebrating the 75th birthday of A A Slavno
A Fibonacci sequence for linear structures with two types of components
We investigate binary voting systems with two types of voters and a hierarchy
among the members in each type, so that members in one class have more
influence or importance than members in the other class. The purpose of this
paper is to count, up to isomorphism, the number of these voting systems for an
arbitrary number of voters. We obtain a closed formula for the number of these
systems, this formula follows a Fibonacci sequence with a smooth polynomial
variation on the number of voters.Comment: All the results contained in this file are included in a paper
submitted to Annals of Operations Research in October, 2008 on ocasion of the
Conference on Applied Mathematical Programming and Modelling, that held in
Bratislava in May, 200
Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data
Constraint Programming (CP) has proved an effective paradigm to model and
solve difficult combinatorial satisfaction and optimisation problems from
disparate domains. Many such problems arising from the commercial world are
permeated by data uncertainty. Existing CP approaches that accommodate
uncertainty are less suited to uncertainty arising due to incomplete and
erroneous data, because they do not build reliable models and solutions
guaranteed to address the user's genuine problem as she perceives it. Other
fields such as reliable computation offer combinations of models and associated
methods to handle these types of uncertain data, but lack an expressive
framework characterising the resolution methodology independently of the model.
We present a unifying framework that extends the CP formalism in both model
and solutions, to tackle ill-defined combinatorial problems with incomplete or
erroneous data. The certainty closure framework brings together modelling and
solving methodologies from different fields into the CP paradigm to provide
reliable and efficient approches for uncertain constraint problems. We
demonstrate the applicability of the framework on a case study in network
diagnosis. We define resolution forms that give generic templates, and their
associated operational semantics, to derive practical solution methods for
reliable solutions.Comment: Revised versio
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