1,239,697 research outputs found

    Finding the maximum and minimum

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    AbstractWe consider the problem of finding the maximum out of a list of n ordered items with binary comparisons where the pth fraction of the answers may be false. It is shown that the maximum can be determined iff p < 12 and that a successful strategy needs Θ(11−p)n questions. A few similar problems are also discussed, including the problem of finding the maximum and minimum simultaneously with lies and in the nuts and bolts model

    Pareto optimality in house allocation problems

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    We study Pareto optimal matchings in the context of house allocation problems. We present an O(\sqrt{n}m) algorithm, based on Gales Top Trading Cycles Method, for finding a maximum cardinality Pareto optimal matching, where n is the number of agents and m is the total length of the preference lists. By contrast, we show that the problem of finding a minimum cardinality Pareto optimal matching is NP-hard, though approximable within a factor of 2. We then show that there exist Pareto optimal matchings of all sizes between a minimum and maximum cardinality Pareto optimal matching. Finally, we introduce the concept of a signature, which allows us to give a characterization, checkable in linear time, of instances that admit a unique Pareto optimal matching

    Optimal General Matchings

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    Given a graph G=(V,E)G=(V,E) and for each vertex v∈Vv \in V a subset B(v)B(v) of the set {0,1,
,dG(v)}\{0,1,\ldots, d_G(v)\}, where dG(v)d_G(v) denotes the degree of vertex vv in the graph GG, a BB-factor of GG is any set F⊆EF \subseteq E such that dF(v)∈B(v)d_F(v) \in B(v) for each vertex vv, where dF(v)d_F(v) denotes the number of edges of FF incident to vv. The general factor problem asks the existence of a BB-factor in a given graph. A set B(v)B(v) is said to have a {\em gap of length} pp if there exists a natural number k∈B(v)k \in B(v) such that k+1,
,k+p∉B(v)k+1, \ldots, k+p \notin B(v) and k+p+1∈B(v)k+p+1 \in B(v). Without any restrictions the general factor problem is NP-complete. However, if no set B(v)B(v) contains a gap of length greater than 11, then the problem can be solved in polynomial time and Cornuejols \cite{Cor} presented an algorithm for finding a BB-factor, if it exists. In this paper we consider a weighted version of the general factor problem, in which each edge has a nonnegative weight and we are interested in finding a BB-factor of maximum (or minimum) weight. In particular, this version comprises the minimum/maximum cardinality variant of the general factor problem, where we want to find a BB-factor having a minimum/maximum number of edges. We present an algorithm for the maximum/minimum weight BB-factor for the case when no set B(v)B(v) contains a gap of length greater than 11. This also yields the first polynomial time algorithm for the maximum/minimum cardinality BB-factor for this case

    Tight Bounds on the Synthesis of 3-bit Reversible Circuits: NFT Library

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    The reversible circuit synthesis problem can be reduced to permutation group. This allows Schreier-Sims Algorithm for the strong generating set-finding problem to be used to find tight bounds on the synthesis of 3-bit reversible circuits using the NFT library. The tight bounds include the maximum and minimum length of 3-bit reversible circuits, the maximum and minimum cost of 3-bit reversible circuits. The analysis shows better results than that found in the literature for the lower bound of the cost. The analysis also shows that there are 1960 universal reversible sub-libraries from the main NFT library.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1101.438

    Results on Binary Linear Codes With Minimum Distance 8 and 10

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    All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact bounds for binary linear codes. Primarily two algorithms considering the dual codes are used, namely extension of dual codes with a proper coordinate, and a fast algorithm for finding a maximum clique in a graph, which is modified to find a maximum set of vectors with the right dependency structure.Comment: Submitted to the IEEE Transactions on Information Theory, May 2010 To be presented at the ACCT 201

    Two-channel linear phase FIR QMF bank minimax design via global nonconvex optimization programming

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    In this correspondence, a two-channel linear phase finite impulse response (FIR) quadrature mirror filter (QMF) bank minimax design problem is formulated as a nonconvex optimization problem so that a weighted sum of the maximum amplitude distortion of the filter bank, the maximum passband ripple magnitude and the maximum stopband ripple magnitude of the prototype filter is minimized subject to specifications on these performances. A modified filled function method is proposed for finding the global minimum of the nonconvex optimization problem. Computer numerical simulations show that our proposed design method is efficient and effective

    Profile-Based Optimal Matchings in the Student-Project Allocation Problem

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    In the Student/Project Allocation problem (spa) we seek to assign students to individual or group projects offered by lecturers. Students provide a list of projects they find acceptable in order of preference. Each student can be assigned to at most one project and there are constraints on the maximum number of students that can be assigned to each project and lecturer. We seek matchings of students to projects that are optimal with respect to profile, which is a vector whose rth component indicates how many students have their rth-choice project. We present an efficient algorithm for finding agreedy maximum matching in the spa context – this is a maximum matching whose profile is lexicographically maximum. We then show how to adapt this algorithm to find a generous maximum matching – this is a matching whose reverse profile is lexicographically minimum. Our algorithms involve finding optimal flows in networks. We demonstrate how this approach can allow for additional constraints, such as lecturer lower quotas, to be handled flexibly
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