5,609 research outputs found

    Enumeration of integral tetrahedra

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    We determine the numbers of integral tetrahedra with diameter dd up to isomorphism for all d≤1000d\le 1000 via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most dd in O(d5)O(d^5) time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral 4×44\times 4 matrices with diameter dd fulfilling the triangle inequalities we derive an exact formula.Comment: 10 pages, 1 figur

    Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

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    The study of graph products is a major research topic and typically concerns the term f(G∗H)f(G*H), e.g., to show that f(G∗H)=f(G)f(H)f(G*H)=f(G)f(H). In this paper, we study graph products in a non-standard form f(R[G∗H]f(R[G*H] where RR is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight n1−ϵn^{1-\epsilon}-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where nn is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming NP≠RPNP\neq RP (the weakest possible assumption). (2) A tight n1/2−ϵn^{1/2-\epsilon} hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where nn denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint kk-cycles for large kk. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]
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