1,172 research outputs found
Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs
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
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
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
Plotkin, Rao, and Smith (SODA'97) showed that any graph with edges and
vertices that excludes as a depth -minor has a
separator of size and that such a separator can be
found in time. A time bound of for
any constant 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 and .
If for an arbitrarily small chosen constant
, 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 ) and running time O(\mbox{poly}(h)(\sqrt\ell
n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when . Our third algorithm has running time
O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when . 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
is fixed and . 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
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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