2,155 research outputs found
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
Let be an -node planar graph. In a visibility representation of ,
each node of is represented by a horizontal line segment such that the line
segments representing any two adjacent nodes of are vertically visible to
each other. In the present paper we give the best known compact visibility
representation of . Given a canonical ordering of the triangulated , our
algorithm draws the graph incrementally in a greedy manner. We show that one of
three canonical orderings obtained from Schnyder's realizer for the
triangulated yields a visibility representation of no wider than
. Our easy-to-implement O(n)-time algorithm bypasses the
complicated subroutines for four-connected components and four-block trees
required by the best previously known algorithm of Kant. Our result provides a
negative answer to Kant's open question about whether is a
worst-case lower bound on the required width. Also, if has no degree-three
(respectively, degree-five) internal node, then our visibility representation
for is no wider than (respectively, ).
Moreover, if is four-connected, then our visibility representation for
is no wider than , matching the best known result of Kant and He. As a
by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem
on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to
appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of
Computer Science (STACS), Berlin, Germany, 200
DMVP: Foremost Waypoint Coverage of Time-Varying Graphs
We consider the Dynamic Map Visitation Problem (DMVP), in which a team of
agents must visit a collection of critical locations as quickly as possible, in
an environment that may change rapidly and unpredictably during the agents'
navigation. We apply recent formulations of time-varying graphs (TVGs) to DMVP,
shedding new light on the computational hierarchy of TVG classes by analyzing them in the
context of graph navigation. We provide hardness results for all three classes,
and for several restricted topologies, we show a separation between the classes
by showing severe inapproximability in , limited approximability
in , and tractability in . We also give topologies in
which DMVP in is fixed parameter tractable, which may serve as a
first step toward fully characterizing the features that make DMVP difficult.Comment: 24 pages. Full version of paper from Proceedings of WG 2014, LNCS,
Springer-Verla
JGraphT -- A Java library for graph data structures and algorithms
Mathematical software and graph-theoretical algorithmic packages to
efficiently model, analyze and query graphs are crucial in an era where
large-scale spatial, societal and economic network data are abundantly
available. One such package is JGraphT, a programming library which contains
very efficient and generic graph data-structures along with a large collection
of state-of-the-art algorithms. The library is written in Java with stability,
interoperability and performance in mind. A distinctive feature of this library
is the ability to model vertices and edges as arbitrary objects, thereby
permitting natural representations of many common networks including
transportation, social and biological networks. Besides classic graph
algorithms such as shortest-paths and spanning-tree algorithms, the library
contains numerous advanced algorithms: graph and subgraph isomorphism; matching
and flow problems; approximation algorithms for NP-hard problems such as
independent set and TSP; and several more exotic algorithms such as Berge graph
detection. Due to its versatility and generic design, JGraphT is currently used
in large-scale commercial, non-commercial and academic research projects. In
this work we describe in detail the design and underlying structure of the
library, and discuss its most important features and algorithms. A
computational study is conducted to evaluate the performance of JGraphT versus
a number of similar libraries. Experiments on a large number of graphs over a
variety of popular algorithms show that JGraphT is highly competitive with
other established libraries such as NetworkX or the BGL.Comment: Major Revisio
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