61,292 research outputs found

    Improving Root Separation Bounds

    Get PDF
    International audienceLet f be a polynomial (or polynomial system) with all simple roots. The root separation of f is the minimum of the pair-wise distances between the complex roots. A root separation bound is a lower bound on the root separation. Finding a root separation bound is a fundamental problem, arising in numerous disciplines. We present two new root separation bounds: one univariate bound, and one multivariate bound. The new bounds improve on the old bounds in two ways: (1) The new bounds are usually significantly bigger (hence better) than the previous bounds. (2) The new bounds scale correctly, unlike the previous bounds. Crucially, the new bounds are not harder to compute than the previous bounds

    Stability of roots of polynomials under linear combinations of derivatives

    Get PDF
    Let T=α0I+α1D+...+αnDnT=\alpha_0 I + \alpha_1 D + ...+\alpha_n D^n, where DD is the differentiation operator and α00\alpha_0\not= 0, and let ff be a square-free polynomial with large minimum root separation. We prove that the roots of TfTf are close to the roots of ff translated by α1/α0-\alpha_1/\alpha_0.Comment: 18 pages, 4 figure

    Computing with Tangles

    Full text link
    Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing "k-connected components" of a graph (though in a twisted way). They play an important role in graph minor theory. An interesting aspect of tangles is that they cannot only be defined for graphs, but more generally for arbitrary connectivity functions (that is, integer-valued submodular and symmetric set functions). However, tangles are difficult to deal with algorithmically. To start with, it is unclear how to represent them, because they are families of separations and as such may be exponentially large. Our first contribution is a data structure for representing and accessing all tangles of a graph up to some fixed order. Using this data structure, we can prove an algorithmic version of a very general structure theorem due to Carmesin, Diestel, Harman and Hundertmark (for graphs) and Hundertmark (for arbitrary connectivity functions) that yields a canonical tree decomposition whose parts correspond to the maximal tangles. (This may be viewed as a generalisation of the decomposition of a graph into its 3-connected components.

    Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

    Get PDF
    We give a fixed-parameter tractable algorithm that, given a parameter kk and two graphs G1,G2G_1,G_2, either concludes that one of these graphs has treewidth at least kk, or determines whether G1G_1 and G2G_2 are isomorphic. The running time of the algorithm on an nn-vertex graph is 2O(k5logk)n52^{O(k^5\log k)}\cdot n^5, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in 2O(k5logk)n52^{O(k^5\log k)}\cdot n^5 time that, for a given graph GG on nn vertices, either concludes that the treewidth of GG is at least kk, or: * finds in an isomorphic-invariant way a graph c(G)\mathfrak{c}(G) that is isomorphic to GG; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes GG together with a tree decomposition of GG of width O(k4)O(k^4). Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for G1G_1 and G2G_2 are equal.Comment: Full version of a paper presented at FOCS 201

    On isolation of singular zeros of multivariate analytic systems

    Full text link
    We give a separation bound for an isolated multiple root xx of a square multivariate analytic system ff satisfying that an operator deduced by adding Df(x)Df(x) and a projection of D2f(x)D^2f(x) in a direction of the kernel of Df(x)Df(x) is invertible. We prove that the deflation process applied on ff and this kind of roots terminates after only one iteration. When xx is only given approximately, we give a numerical criterion for isolating a cluster of zeros of ff near xx. We also propose a lower bound of the number of roots in the cluster.Comment: 17 page

    Exact Algorithms for Solving Stochastic Games

    Full text link
    Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games
    corecore