1,689 research outputs found
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths
Computing all-pairs shortest paths is a fundamental and much-studied problem
with many applications. Unfortunately, despite intense study, there are still
no significantly faster algorithms for it than the time
algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist
for the vertex-weighted version if fast matrix multiplication may be used.
Yuster (SODA 2009) gave an algorithm running in time ,
but no combinatorial, truly subcubic algorithm is known.
Motivated by the recent framework of efficient parameterized algorithms (or
"FPT in P"), we investigate the influence of the graph parameters clique-width
() and modular-width () on the running times of algorithms for solving
All-Pairs Shortest Paths. We obtain efficient (and combinatorial) parameterized
algorithms on non-negative vertex-weighted graphs of times
, resp. . If fast matrix
multiplication is allowed then the latter can be improved to
using the algorithm of Yuster as a black box.
The algorithm relative to modular-width is adaptive, meaning that the running
time matches the best unparameterized algorithm for parameter value equal
to , and they outperform them already for for any
On String Graph Limits and the Structure of a Typical String Graph
We study limits of convergent sequences of string graphs, that is, graphs
with an intersection representation consisting of curves in the plane. We use
these results to study the limiting behavior of a sequence of random string
graphs. We also prove similar results for several related graph classes.Comment: 18 page
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
enjoy similar properties as their Euclidean counterparts, but in one dimension
lower: many standard graph problems, such as Independent Set, Dominating Set,
Steiner Tree, and Hamiltonian Cycle can be solved in
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require time under ETH in constant maximum degree
Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in
noisy uniform disk graphs with a matching lower bound
under ETH. This shows that the hyperbolic plane is a potential source of
NP-intermediate problems.Comment: Short version appears in SODA 202
Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope
We study a new geometric graph parameter \egd(G), defined as the smallest
integer for which any partial symmetric matrix which is completable to
a correlation matrix and whose entries are specified at the positions of the
edges of , can be completed to a matrix in the convex hull of correlation
matrices of \rank at most . This graph parameter is motivated by its
relevance to the problem of finding low rank solutions to semidefinite programs
over the elliptope, and also by its relevance to the bounded rank Grothendieck
constant. Indeed, \egd(G)\le r if and only if the rank- Grothendieck
constant of is equal to 1. We show that the parameter \egd(G) is minor
monotone, we identify several classes of forbidden minors for \egd(G)\le r
and we give the full characterization for the case . We also show an upper
bound for \egd(G) in terms of a new tree-width-like parameter \sla(G),
defined as the smallest for which is a minor of the strong product of a
tree and . We show that, for any 2-connected graph on at
least 6 nodes, \egd(G)\le 2 if and only if \sla(G)\le 2.Comment: 33 pages, 8 Figures. In its second version, the paper has been
modified to accommodate the suggestions of the referees. Furthermore, the
title has been changed since we feel that the new title reflects more
accurately the content and the main results of the pape
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