189 research outputs found
An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL
In general, a graph modification problem is defined by a graph modification
operation and a target graph property . Typically, the
modification operation may be vertex removal}, edge removal}, edge
contraction}, or edge addition and the question is, given a graph and an
integer , whether it is possible to transform to a graph in
after applying times the operation on . This problem has
been extensively studied for particilar instantiations of and
. In this paper we consider the general property
of being planar and, moreover, being a model of some First-Order Logic sentence
(an FOL-sentence). We call the corresponding meta-problem Graph
-Modification to Planarity and and prove the following
algorithmic meta-theorem: there exists a function
such that, for every and every FOL sentence , the Graph
-Modification to Planarity and is solvable in
time. The proof constitutes a hybrid of two different
classic techniques in graph algorithms. The first is the irrelevant vertex
technique that is typically used in the context of Graph Minors and deals with
properties such as planarity or surface-embeddability (that are not
FOL-expressible) and the second is the use of Gaifman's Locality Theorem that
is the theoretical base for the meta-algorithmic study of FOL-expressible
problems
Fixed-Parameter Tractability of Maximum Colored Path and Beyond
We introduce a general method for obtaining fixed-parameter algorithms for
problems about finding paths in undirected graphs, where the length of the path
could be unbounded in the parameter. The first application of our method is as
follows.
We give a randomized algorithm, that given a colored -vertex undirected
graph, vertices and , and an integer , finds an -path
containing at least different colors in time . This is the
first FPT algorithm for this problem, and it generalizes the algorithm of
Bj\"orklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through
specified vertices. It also implies the first time algorithm for
finding an -path of length at least .
Our method yields FPT algorithms for even more general problems. For example,
we consider the problem where the input consists of an -vertex undirected
graph , a matroid whose elements correspond to the vertices of and
which is represented over a finite field of order , a positive integer
weight function on the vertices of , two sets of vertices , and integers , and the task is to find vertex-disjoint paths
from to so that the union of the vertices of these paths contains an
independent set of of cardinality and weight , while minimizing the
sum of the lengths of the paths. We give a
time randomized algorithm for this problem.Comment: 50 pages, 16 figure
Shortest Cycles With Monotone Submodular Costs
We introduce the following submodular generalization of the Shortest Cycle
problem. For a nonnegative monotone submodular cost function defined on the
edges (or the vertices) of an undirected graph , we seek for a cycle in
of minimum cost . We give an algorithm that given an
-vertex graph , parameter , and the function
represented by an oracle, in time finds a
cycle in with . This is in
sharp contrast with the non-approximability of the closely related Monotone
Submodular Shortest -Path problem, which requires exponentially many
queries to the oracle for finding an -approximation [Goel
et al., FOCS 2009]. We complement our algorithm with a matching lower bound. We
show that for every , obtaining a
-approximation requires at least queries to the oracle. When the function is integer-valued,
our algorithm yields that a cycle of cost can be found in time
. In particular, for
this gives a quasipolynomial-time algorithm
computing a cycle of minimum submodular cost. Interestingly, while a
quasipolynomial-time algorithm often serves as a good indication that a
polynomial time complexity could be achieved, we show a lower bound that
queries are required even when .Comment: 17 pages, 1 figure. Accepted to SODA 202
Compound Logics for Modification Problems
We introduce a novel model-theoretic framework inspired from graph
modification and based on the interplay between model theory and algorithmic
graph minors. The core of our framework is a new compound logic operating with
two types of sentences, expressing graph modification: the modulator sentence,
defining some property of the modified part of the graph, and the target
sentence, defining some property of the resulting graph. In our framework,
modulator sentences are in counting monadic second-order logic (CMSOL) and have
models of bounded treewidth, while target sentences express first-order logic
(FOL) properties along with minor-exclusion. Our logic captures problems that
are not definable in first-order logic and, moreover, may have instances of
unbounded treewidth. Also, it permits the modeling of wide families of problems
involving vertex/edge removals, alternative modulator measures (such as
elimination distance or -treewidth), multistage modifications, and
various cut problems. Our main result is that, for this compound logic,
model-checking can be done in quadratic time. All derived algorithms are
constructive and this, as a byproduct, extends the constructibility horizon of
the algorithmic applications of the Graph Minors theorem of Robertson and
Seymour. The proposed logic can be seen as a general framework to capitalize on
the potential of the irrelevant vertex technique. It gives a way to deal with
problem instances of unbounded treewidth, for which Courcelle's theorem does
not apply. The proof of our meta-theorem combines novel combinatorial results
related to the Flat Wall theorem along with elements of the proof of
Courcelle's theorem and Gaifman's theorem. We finally prove extensions where
the target property is expressible in FOL+DP, i.e., the enhancement of FOL with
disjoint-paths predicates
Combining fish and benthic communities into multiple regimes reveals complex reef dynamics
Abstract Coral reefs worldwide face an uncertain future with many reefs reported to transition from being dominated by corals to macroalgae. However, given the complexity and diversity of the ecosystem, research on how regimes vary spatially and temporally is needed. Reef regimes are most often characterised by their benthic components; however, complex dynamics are associated with losses and gains in both fish and benthic assemblages. To capture this complexity, we synthesised 3,345 surveys from Hawai‘i to define reef regimes in terms of both fish and benthic assemblages. Model-based clustering revealed five distinct regimes that varied ecologically, and were spatially heterogeneous by island, depth and exposure. We identified a regime characteristic of a degraded state with low coral cover and fish biomass, one that had low coral but high fish biomass, as well as three other regimes that varied significantly in their ecology but were previously considered a single coral dominated regime. Analyses of time series data reflected complex system dynamics, with multiple transitions among regimes that were a function of both local and global stressors. Coupling fish and benthic communities into reef regimes to capture complex dynamics holds promise for monitoring reef change and guiding ecosystem-based management of coral reefs
Neutrino and Antineutrino Inclusive Charged-current Cross Section Measurements with the MINOS Near Detector
The energy dependence of the neutrino-iron and antineutrino-iron inclusive
charged-current cross sections and their ratio have been measured using a
high-statistics sample with the MINOS Near Detector exposed to the NuMI beam
from the Main Injector at Fermilab. Neutrino and antineutrino fluxes were
determined using a low hadronic energy subsample of charged-current events. We
report measurements of neutrino-Fe (antineutrinoFe) cross section in the energy
range 3-50 GeV (5-50 GeV) with precision of 2-8% (3-9%) and their ratio which
is measured with precision 2-8%. The data set spans the region from low energy,
where accurate measurements are sparse, up to the high-energy scaling region
where the cross section is well understood.Comment: accepted by PR
A Study of Muon Neutrino Disappearance Using the Fermilab Main Injector Neutrino Beam
We report the results of a search for muon-neutrino disappearance by the Main
Injector Neutrino Oscillation Search. The experiment uses two detectors
separated by 734 km to observe a beam of neutrinos created by the Neutrinos at
the Main Injector facility at Fermi National Accelerator Laboratory. The data
were collected in the first 282 days of beam operations and correspond to an
exposure of 1.27e20 protons on target. Based on measurements in the Near
Detector, in the absence of neutrino oscillations we expected 336 +/- 14
muon-neutrino charged-current interactions at the Far Detector but observed
215. This deficit of events corresponds to a significance of 5.2 standard
deviations. The deficit is energy dependent and is consistent with two-flavor
neutrino oscillations according to delta m-squared = 2.74e-3 +0.44/-0.26e-3
eV^2 and sin^2(2 theta) > 0.87 at 68% confidence level.Comment: In submission to Phys. Rev.
Measurement of neutrino velocity with the MINOS detectors and NuMI neutrino beam
The velocity of a ~3 GeV neutrino beam is measured by comparing detection times at the near and far detectors of the MINOS experiment, separated by 734 km. A total of 473 far detector neutrino events was used to measure (v-c)/c=5.12.910-5 (at 68% C.L.). By correlating the measured energies of 258 charged-current neutrino events to their arrival times at the far detector, a limit is imposed on the neutrino mass of mnu<50 MeV/c2 (99% C.L.)
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