An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

In general, a graph modification problem is defined by a graph modification operation $\boxtimes$ and a target graph property ${\cal P}$. Typically, the modification operation $\boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to transform $G$ to a graph in ${\cal P}$ after applying $k$ times the operation $\boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $\boxtimes$ and ${\cal P}$. In this paper we consider the general property ${\cal P}_{{\phi}}$ of being planar and, moreover, being a model of some First-Order Logic sentence ${\phi}$ (an FOL-sentence). We call the corresponding meta-problem Graph $\boxtimes$-Modification to Planarity and ${\phi}$ and prove the following algorithmic meta-theorem: there exists a function $f:\Bbb{N}^{2}\to\Bbb{N}$ such that, for every $\boxtimes$ and every FOL sentence ${\phi}$, the Graph $\boxtimes$-Modification to Planarity and ${\phi}$ is solvable in $f(k,|{\phi}|)\cdot n^2$ 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 $n$-vertex undirected graph, vertices $s$ and $t$, and an integer $k$, finds an $(s,t)$-path containing at least $k$ different colors in time $2^k n^{O(1)}$. 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 $k$ specified vertices. It also implies the first $2^k n^{O(1)}$ time algorithm for finding an $(s,t)$-path of length at least $k$. Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an $n$-vertex undirected graph $G$, a matroid $M$ whose elements correspond to the vertices of $G$ and which is represented over a finite field of order $q$, a positive integer weight function on the vertices of $G$, two sets of vertices $S,T \subseteq V(G)$, and integers $p,k,w$, and the task is to find $p$ vertex-disjoint paths from $S$ to $T$ so that the union of the vertices of these paths contains an independent set of $M$ of cardinality $k$ and weight $w$, while minimizing the sum of the lengths of the paths. We give a $2^{p+O(k^2 \log (q+k))} n^{O(1)} w$ 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 $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of minimum cost $\textsf{OPT}=f(C)$. We give an algorithm that given an $n$-vertex graph $G$, parameter $\varepsilon > 0$, and the function $f$ represented by an oracle, in time $n^{\mathcal{O}(\log 1/\varepsilon)}$ finds a cycle $C$ in $G$ with $f(C)\leq (1+\varepsilon)\cdot \textsf{OPT}$. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest $(s,t)$-Path problem, which requires exponentially many queries to the oracle for finding an $n^{2/3-\varepsilon}$-approximation [Goel et al., FOCS 2009]. We complement our algorithm with a matching lower bound. We show that for every $\varepsilon > 0$, obtaining a $(1+\varepsilon)$-approximation requires at least $n^{\Omega(\log 1/ \varepsilon)}$ queries to the oracle. When the function $f$ is integer-valued, our algorithm yields that a cycle of cost $\textsf{OPT}$ can be found in time $n^{\mathcal{O}(\log \textsf{OPT})}$. In particular, for $\textsf{OPT}=n^{\mathcal{O}(1)}$ 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 $n^{\mathcal{O}(\log n)}$ queries are required even when $\textsf{OPT} = \mathcal{O}(n)$.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 $\mathcal{G}$-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.)