23 research outputs found

    Voting for Committees in Agreeable Societies

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    We examine the following voting situation. A committee of kk people is to be formed from a pool of n candidates. The voters selecting the committee will submit a list of jj candidates that they would prefer to be on the committee. We assume that jk<nj \leq k < n. For a chosen committee, a given voter is said to be satisfied by that committee if her submitted list of jj candidates is a subset of that committee. We examine how popular is the most popular committee. In particular, we show there is always a committee that satisfies a certain fraction of the voters and examine what characteristics of the voter data will increase that fraction.Comment: 11 pages; to appear in Contemporary Mathematic

    Voter Compatibility In Interval Societies

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    In an interval society, voters are represented by intervals on the real line, corresponding to their approval sets on a linear political spectrum. I imagine the society to be a representative democracy, and ask how to choose members of the society as representatives. Following work in mathematical psychology by Coombs and others, I develop a measure of the compatibility (political similarity) of two voters. I use this measure to determine the popularity of each voter as a candidate. I then establish local “agreeability” conditions and attempt to find a lower bound for the popularity of the best candidate. Other results about certain special societies are also obtaine

    Interval voting technique for active and dormant state synchronization of the nodes in WSN

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    Energy conservation is one of the main issues of WSN research. To save energy and prolong sensor nodes lives, nodes operate in a duty-cycling work mode. Synchronous duty-cycling protocols require time synchronization in order to enable nodes to simultaneously sleep and wake up. In this paper, a method for nodes synchronization based on interval voting is proposed. The method allows to obtain consensus relation in a form of interval by means of sensor data fusion technique and apply it for nodes synchronization, thus solving a problem of choosing reference time node

    On (2,3)-agreeable Box Societies

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    The notion of (k,m)(k,m)-agreeable society was introduced by Deborah Berg et al.: a family of convex subsets of Rd\R^d is called (k,m)(k,m)-agreeable if any subfamily of size mm contains at least one non-empty kk-fold intersection. In that paper, the (k,m)(k,m)-agreeability of a convex family was shown to imply the existence of a subfamily of size βn\beta n with non-empty intersection, where nn is the size of the original family and β[0,1]\beta\in[0,1] is an explicit constant depending only on k,mk,m and dd. The quantity β(k,m,d)\beta(k,m,d) is called the minimal \emph{agreement proportion} for a (k,m)(k,m)-agreeable family in Rd\R^d. If we only assume that the sets are convex, simple examples show that β=0\beta=0 for (k,m)(k,m)-agreeable families in Rd\R^d where k<dk<d. In this paper, we introduce new techniques to find positive lower bounds when restricting our attention to families of dd-boxes, i.e. cuboids with sides parallel to the coordinates hyperplanes. We derive explicit formulas for the first non-trivial case: the case of (2,3)(2,3)-agreeable families of dd-boxes with d2d\geq 2.Comment: 15 pages, 10 figure

    Schedule (2007)

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    Sixteenth Conference of the Association of Christians in the Mathematical Science

    Tur\'an and Ramsey Properties of Subcube Intersection Graphs

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    The discrete cube {0,1}d\{0,1\}^d is a fundamental combinatorial structure. A subcube of {0,1}d\{0,1\}^d is a subset of 2k2^k of its points formed by fixing kk coordinates and allowing the remaining dkd-k to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no r+1r+1 of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no kk which have non-empty intersection and no ll which are pairwise disjoint? These questions are naturally expressed as Tur\'an and Ramsey type questions in intersection graphs of subcubes where the intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Tur\'an and Ramsey type problems are at the heart of extremal combinatorics and so these problems are mathematically natural. However, a second motivation is a connection with some questions in social choice theory arising from a simple model of agreement in a society. Specifically, if we have to make a binary choice on each of nn separate issues then it is reasonable to assume that the set of choices which are acceptable to an individual will be represented by a subcube. Consequently, the pattern of intersections within a family of subcubes will have implications for the level of agreement within a society. We pose a number of questions and conjectures relating directly to the Tur\'an and Ramsey problems as well as raising some further directions for study of subcube intersection graphs.Comment: 18 page

    Random subcube intersection graphs I: cliques and covering

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    We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube QdQ_d to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their intersection is non-empty. Our motivation for considering such graphs is to model `random compatibility' between vertices in a large network. For both of the models considered in this paper, we determine the thresholds for covering the underlying hypercube QdQ_d and for the appearance of s-cliques. In addition we pose some open problems.Comment: 38 pages, 1 figur
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