147 research outputs found

    Polytopes associated to Dihedral Groups

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    In this note we investigate the convex hull of those n×nn \times n-permutation matrices that correspond to symmetries of a regular nn-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart hh^*-vector

    Explicit Non-Adaptive Combinatorial Group Testing Schemes

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    Group testing is a long studied problem in combinatorics: A small set of rr ill people should be identified out of the whole (nn people) by using only queries (tests) of the form "Does set X contain an ill human?". In this paper we provide an explicit construction of a testing scheme which is better (smaller) than any known explicit construction. This scheme has \bigT{\min[r^2 \ln n,n]} tests which is as many as the best non-explicit schemes have. In our construction we use a fact that may have a value by its own right: Linear error-correction codes with parameters [m,k,δm]q[m,k,\delta m]_q meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in \bigT{q^km} time.Comment: 15 pages, accepted to ICALP 200

    Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

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    The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

    Higher-order CIS codes

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    We introduce {\bf complementary information set codes} of higher-order. A binary linear code of length tktk and dimension kk is called a complementary information set code of order tt (tt-CIS code for short) if it has tt pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a tt-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order 33 is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length 12\le 12 is given. Nonlinear codes better than linear codes are derived by taking binary images of Z4\Z_4-codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate 1/t1/t it either provides tt disjoint information sets or proves that the code is not tt-CIS. Using this algorithm, all optimal or best known [tk,k][tk, k] codes where t=3,4,,256t=3, 4, \dots, 256 and 1k256/t1 \le k \le \lfloor 256/t \rfloor are shown to be tt-CIS for all such kk and tt, except for t=3t=3 with k=44k=44 and t=4t=4 with k=37k=37.Comment: 13 pages; 1 figur

    A Tutorial on Clique Problems in Communications and Signal Processing

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    Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and kk-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems

    CWI-evaluation - Progress Report 1993-1998

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    Dagstuhl News January - December 2001

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    "Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

    Sign rank versus VC dimension

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    This work studies the maximum possible sign rank of N×NN \times N sign matrices with a given VC dimension dd. For d=1d=1, this maximum is {three}. For d=2d=2, this maximum is Θ~(N1/2)\tilde{\Theta}(N^{1/2}). For d>2d >2, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an O(N/log(N))O(N/\log(N)) multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the N×NN \times N adjacency matrix of a Δ\Delta regular graph with a second eigenvalue of absolute value λ\lambda and ΔN/2\Delta \leq N/2. We show that the sign rank of the signed version of this matrix is at least Δ/λ\Delta/\lambda. We use this connection to prove the existence of a maximum class C{±1}NC\subseteq\{\pm 1\}^N with VC dimension 22 and sign rank Θ~(N1/2)\tilde{\Theta}(N^{1/2}). This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC dimension". Additional results in this version: (i) Estimates on the number of maximum VC classes (answering a question of Frankl from '89). (ii) Estimates on the sign rank of large VC classes (answering a question of Ben-David et al. from '03). (iii) A discussion on the computational complexity of computing the sign-ran
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