4,574 research outputs found

    Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.

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    We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time

    Lossy Kernelization

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    In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size α\alpha-approximate kernel. Loosely speaking, a polynomial size α\alpha-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance (I,k)(I,k) to a parameterized problem, and outputs another instance (I′,k′)(I',k') to the same problem, such that ∣I′∣+k′≤kO(1)|I'|+k' \leq k^{O(1)}. Additionally, for every c≥1c \geq 1, a cc-approximate solution s′s' to the pre-processed instance (I′,k′)(I',k') can be turned in polynomial time into a (c⋅α)(c \cdot \alpha)-approximate solution ss to the original instance (I,k)(I,k). Our main technical contribution are α\alpha-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless NP⊆coNP/polyNP \subseteq coNP/poly. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an α\alpha-approximate kernel of polynomial size, for any α≥1\alpha \geq 1, unless NP⊆coNP/polyNP \subseteq coNP/poly. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and approximate kernel lower bounds for Set Cover and Hitting Set parameterized by universe siz

    Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits

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    The Nielsen-Thurston theory of surface diffeomorphisms shows that useful dynamical information can be obtained about a surface diffeomorphism from a finite collection of periodic orbits.In this paper, we extend these results to homoclinic and heteroclinic orbits of saddle points. These orbits are most readily computed and studied as intersections of unstable and stable manifolds comprising homoclinic or heteroclinic tangles in the surface. We show how to compute a map of a one-dimensional space similar to a train-track which represents the isotopy-stable dynamics of the surface diffeomorphism relative to a tangle. All orbits of this one-dimensional representative are globally shadowed by orbits of the surface diffeomorphism, and periodic, homoclinic and heteroclinic orbits of the one-dimensional representative are shadowed by similar orbits in the surface.By constructing suitable surface diffeomorphisms, we prove that these results are optimal in the sense that the topological entropy of the one-dimensional representative is the greatest lower bound for the entropies of diffeomorphisms in the isotopy class.Comment: Version submitted to "Dynamical Systems: An International Journal" Section 7 has been further revised; the method for pA maps is new. Notation has been standardised throughou

    The k-fixed-endpoint path partition problem

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    The Hamiltonian path problem is to determine whether a graph has a Hamiltonian path. This problem is NP-complete in general. The path partition problem is to determine the minimum number of vertex-disjoint paths required to cover a graph. Since this problem is a generalization of the Hamiltonian path problem, it is also NP-complete in general. The k-fixed-endpoint path partition problem is to determine the minimum number of vertex-disjoint paths required to cover a graphG such that each vertex in a set T of k vertices is an endpoint of a path. Since this problem is a generalization of the Hamiltonian path problem and path partition problem, it is also NP-complete in general. For certain classes of graphs, there exist efficient algorithms for the k-fixed-endpoint path partition problem. We consider this problem restricted to trees, threshold graphs, block graphs, and unit interval graphs and show min-max theorems which characterize the k-fixed-endpoint pathpartition number
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