100 research outputs found
Algorithms for flows and disjoint paths in planar graphs
In this dissertation we describe several algorithms for computing flows, connectivity, and disjoint paths in planar graphs. In all cases, the algorithms are either the first polynomial-time algorithms or are faster than all previously-known algorithms.
First, we describe algorithms for the maximum flow problem in directed planar graphs with integer capacities on both vertices and arcs and with multiple sources and sinks. The algorithms are the first to solve the problem in near-linear time when the number of terminals is fixed and the capacities are polynomially bounded. As a byproduct, we get the first algorithm to solve the vertex-disjoint S-T paths problem in near-linear time when the number of terminals is fixed but greater than 2. We also modify our algorithms to handle real capacities in near-linear time when they are three terminals.
Second, we describe algorithms to compute element-connectivity and a related structure called the reduced graph. We show that global element-connectivity in planar graphs can be found in linear time if the terminals can be covered by O(1) faces. We also show that the reduced graph can be computed in subquadratic time in planar graphs if the number of terminals is fixed.
Third, we describe algorithms for solving or approximately solving the vertex-disjoint paths problem when we want to minimize the total length of the paths. For planar graphs, we describe: (1) an exact algorithm for the case of four pairs of terminals on a single face; and (2) a k-approximation algorithm for the case of k pairs of terminals on a single face.
Fourth, we describe algorithms and a hardness result for the ideal orientation problem. We show that the problem is NP-hard in planar graphs. On the other hand, we show that the problem is polynomial-time solvable in planar graphs when the number of terminals is fixed, the terminals are all on the same face, and no two of the terminal pairs cross. We also describe an algorithm for serial instances of a generalization of the ideal orientation problem called the k-min-sum orientation problem
Topologically Trivial Closed Walks in Directed Surface Graphs
Let be a directed graph with vertices and edges, embedded on a
surface , possibly with boundary, with first Betti number . We
consider the complexity of finding closed directed walks in that are either
contractible (trivial in homotopy) or bounding (trivial in integer homology) in
. Specifically, we describe algorithms to determine whether contains a
simple contractible cycle in time, or a contractible closed walk in
time, or a bounding closed walk in time. Our
algorithms rely on subtle relationships between strong connectivity in and
in the dual graph ; our contractible-closed-walk algorithm also relies on
a seminal topological result of Hass and Scott. We also prove that detecting
simple bounding cycles is NP-hard.
We also describe three polynomial-time algorithms to compute shortest
contractible closed walks, depending on whether the fundamental group of the
surface is free, abelian, or hyperbolic. A key step in our algorithm for
hyperbolic surfaces is the construction of a context-free grammar with
non-terminals that generates all contractible closed walks of
length at most L, and only contractible closed walks, in a system of quads of
genus . Finally, we show that computing shortest simple contractible
cycles, shortest simple bounding cycles, and shortest bounding closed walks are
all NP-hard.Comment: 30 pages, 18 figures; fixed several minor bugs and added one figure.
An extended abstraction of this paper will appear at SOCG 201
Epistemic Syllogistic: First Steps
Aristotle's discussions on modal syllogistic have often been viewed as
error-prone and have garnered significant attention in the literature due to
historical and philosophical interests. However, from a contemporary
standpoint, they also introduced natural fragments of first-order modal logic,
warranting a comprehensive technical analysis. In this paper, drawing
inspiration from the natural logic program, we propose and examine several
variants of modal syllogistic within the epistemic context, thereby coining the
term Epistemic Syllogistic. Specifically, we concentrate on the de re
interpretation of epistemic syllogisms containing non-trivial yet natural
expressions such as "all things known to be A are also known to be not B." We
explore the epistemic apodeictic syllogistic and its extensions, which
accommodate more complex terms. Our main contributions include several
axiomatizations of these logics, with completeness proofs that may be of
independent interest.Comment: In Proceedings TARK 2023, arXiv:2307.0400
Topologically Trivial Closed Walks in Directed Surface Graphs
Let G be a directed graph with n vertices and m edges, embedded on a surface S, possibly with boundary, with first Betti number beta. We consider the complexity of finding closed directed walks in G that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in S. Specifically, we describe algorithms to determine whether G contains a simple contractible cycle in O(n+m) time, or a contractible closed walk in O(n+m) time, or a bounding closed walk in O(beta (n+m)) time. Our algorithms rely on subtle relationships between strong connectivity in G and in the dual graph G^*; our contractible-closed-walk algorithm also relies on a seminal topological result of Hass and Scott. We also prove that detecting simple bounding cycles is NP-hard.
We also describe three polynomial-time algorithms to compute shortest contractible closed walks, depending on whether the fundamental group of the surface is free, abelian, or hyperbolic. A key step in our algorithm for hyperbolic surfaces is the construction of a context-free grammar with O(g^2L^2) non-terminals that generates all contractible closed walks of length at most L, and only contractible closed walks, in a system of quads of genus g >= 2. Finally, we show that computing shortest simple contractible cycles, shortest simple bounding cycles, and shortest bounding closed walks are all NP-hard
Estimation of Vegetation Latent Heat Flux over Three Forest Sites in ChinaFLUX using Satellite Microwave Vegetation Water Content Index
Latent heat flux (LE) and the corresponding water vapor lost from the Earth's surface to the atmosphere, which is called Evapotranspiration (ET), is one of the key processes in the water cycle and energy balance of the global climate system. Satellite remote sensing is the only feasible technique to estimate LE over a large-scale region. While most of the previous satellite LE methods are based on the optical vegetation index (VI), here we propose a microwave-VI (EDVI) based LE algorithm which can work for both day and night time, and under clear or non-raining conditions. This algorithm is totally driven by multiple-sensor satellite products of vegetation water content index, solar radiation, and cloud properties, with some aid from a reanalysis dataset. The satellite inputs and the performance of this algorithm are validated with in situ measurements at three ChinaFLUX forest sites. Our results show that the selected satellite observations can indeed serve as the inputs for the purpose of estimating ET. The instantaneous estimations of LE (LEcal) from this algorithm show strong positive temporal correlations with the in situ measured LE (LEobs) with the correlation coefficients (R) of 0.56-0.88 in the study years. The mean bias is kept within 16.0% (23.0W/m2) across the three sites. At the monthly scale, the correlations between the retrieval and the in situ measurements are further improved to an R of 0.84-0.95 and the bias is less than 14.3%. The validation results also indicate that EDVI-based LE method can produce stable LEcal under different cloudy skies with good accuracy. Being independent of any in situ measurements as inputs, this algorithm shows great potential for estimating ET under both clear and cloudy skies on a global scale for climate study
Tutorial: Nonlinear magnonics
Nonlinear magnonics studies the nonlinear interaction between magnons and
other physical platforms (phonon, photon, qubit, spin texture) to generate
novel magnon states for information processing. In this tutorial, we first
introduce the nonlinear interactions of magnons in pure magnetic systems and
hybrid magnon-phonon and magnon-photon systems. Then we show how these
nonlinear interactions can generate exotic magnonic phenomena. In the classical
regime, we will cover the parametric excitation of magnons, bistability and
multistability, and the magnonic frequency comb. In the quantum regime, we will
discuss the single magnon state, Schr\"{o}dinger cat state and the entanglement
and quantum steering among magnons, photons and phonons. The applications of
the hybrid magnonics systems in quantum transducer and sensing will also be
presented. Finally, we outlook the future development direction of nonlinear
magnonics.Comment: 50 pages, 26 figure
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