1,172 research outputs found

    Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs

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    A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of \epsilon and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(n lg^2 n). However, our constructions have slower query times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our linear-space results, we can in fact ensure, for any delta > 0, that the space required is only 1 + delta times the space required just to represent the graph itself

    Local tree-width, excluded minors, and approximation algorithms

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    The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with excluded minors that essentially says that such graphs can be decomposed into trees of graphs of bounded local tree-width. As an application of this theorem, we show that a number of combinatorial optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Independent Set have a polynomial time approximation scheme when restricted to a class of graphs with an excluded minor

    Simple PTAS's for families of graphs excluding a minor

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    We show that very simple algorithms based on local search are polynomial-time approximation schemes for Maximum Independent Set, Minimum Vertex Cover and Minimum Dominating Set, when the input graphs have a fixed forbidden minor.Comment: To appear in Discrete Applied Mathematic

    Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles

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    Plotkin, Rao, and Smith (SODA'97) showed that any graph with mm edges and nn vertices that excludes KhK_h as a depth O(logn)O(\ell\log n)-minor has a separator of size O(n/+h2logn)O(n/\ell + \ell h^2\log n) and that such a separator can be found in O(mn/)O(mn/\ell) time. A time bound of O(m+n2+ϵ/)O(m + n^{2+\epsilon}/\ell) for any constant ϵ>0\epsilon > 0 was later given (W., FOCS'11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has the same separator size and running time O(\mbox{poly}(h)\ell m^{1+\epsilon}). This is a significant improvement for small hh and \ell. If =Ω(nϵ)\ell = \Omega(n^{\epsilon'}) for an arbitrarily small chosen constant ϵ>0\epsilon' > 0, we get a time bound of O(\mbox{poly}(h)\ell n^{1+\epsilon}). The second algorithm achieves the same separator size (with a slightly larger polynomial dependency on hh) and running time O(\mbox{poly}(h)(\sqrt\ell n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when =Ω(nϵ)\ell = \Omega(n^{\epsilon'}). Our third algorithm has running time O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when =Ω(nϵ)\ell = \Omega(n^{\epsilon'}). It finds a separator of size O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n) which is no worse than previous bounds when hh is fixed and =O~(n1/4)\ell = \tilde O(n^{1/4}). A main tool in obtaining our results is a novel application of a decremental approximate distance oracle of Roditty and Zwick.Comment: 16 pages. Full version of the paper that appeared at ICALP'14. Minor fixes regarding the time bounds such that these bounds hold also for non-sparse graph

    Grad and Classes with Bounded Expansion II. Algorithmic Aspects

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    Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). These classes are also characterized by the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. These results lead to several new linear time algorithms, such as an algorithm for counting all the isomorphs of a fixed graph in an input graph or an algorithm for checking whether there exists a subset of vertices of a priori bounded size such that the subgraph induced by this subset satisfies some arbirtrary but fixed first order sentence. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. We also show, extending several earlier results, that a class of graphs has sublinear separators if it has sub-exponential expansion. This result result is best possible in general
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