35,661 research outputs found
Min-Cost Flow in Unit-Capacity Planar Graphs
In this paper we give an O~((nm)^(2/3) log C) time algorithm for computing min-cost flow (or min-cost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by C. For planar multigraphs, this improves upon the best known algorithms for general graphs: the O~(m^(10/7) log C) time algorithm of Cohen et al. [SODA 2017], the O(m^(3/2) log(nC)) time algorithm of Gabow and Tarjan [SIAM J. Comput. 1989] and the O~(sqrt(n) m log C) time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the Omega(m^(3/2)) time barrier for min-cost flow problem in planar graphs.
To obtain our result we first give a very simple successive shortest paths based scaling algorithm for unit-capacity min-cost flow problem that does not explicitly operate on dual variables. This algorithm also runs in O~(m^(3/2) log C) time for general graphs, and, to the best of our knowledge, it has not been described before. We subsequently show how to implement this algorithm faster on planar graphs using well-established tools: r-divisions and efficient algorithms for computing (shortest) paths in so-called dense distance graphs
Maximum Edge-Disjoint Paths in -sums of Graphs
We consider the approximability of the maximum edge-disjoint paths problem
(MEDP) in undirected graphs, and in particular, the integrality gap of the
natural multicommodity flow based relaxation for it. The integrality gap is
known to be even for planar graphs due to a simple
topological obstruction and a major focus, following earlier work, has been
understanding the gap if some constant congestion is allowed.
In this context, it is natural to ask for which classes of graphs does a
constant-factor constant-congestion property hold. It is easy to deduce that
for given constant bounds on the approximation and congestion, the class of
"nice" graphs is nor-closed. Is the converse true? Does every proper
minor-closed family of graphs exhibit a constant factor, constant congestion
bound relative to the LP relaxation? We conjecture that the answer is yes.
One stumbling block has been that such bounds were not known for bounded
treewidth graphs (or even treewidth 3). In this paper we give a polytime
algorithm which takes a fractional routing solution in a graph of bounded
treewidth and is able to integrally route a constant fraction of the LP
solution's value. Note that we do not incur any edge congestion. Previously
this was not known even for series parallel graphs which have treewidth 2. The
algorithm is based on a more general argument that applies to -sums of
graphs in some graph family, as long as the graph family has a constant factor,
constant congestion bound. We then use this to show that such bounds hold for
the class of -sums of bounded genus graphs
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
Microscopic self-consistent theory of Josephson junctions including dynamical electron correlations
We formulate a fully self-consistent, microscopic model to study the
retardation and correlation effects of the barrier within a Josephson junction.
The junction is described by a series of planes, with electronic correlation
included through a local self energy for each plane. We calculate current-phase
relationships for various junctions, which include non-magnetic impurities in
the barrier region, or an interfacial scattering potential. Our results
indicate that the linear response of the supercurrent to phase across the
barrier region is a good, but not exact indicator of the critical current. Our
calculations of the local density of states show the current-carrying Andreev
bound states and their energy evolution with the phase difference across the
junction.
We calculate the figure of merit for a Josephson junction, which is the
product of the critical current, Ic, and the normal state resistance, R(N), for
junctions with different barrier materials. The normal state resistance is
calculated using the Kubo formula, for a system with zero current flow and no
superconducting order. Semiclassical calculations would predict that these two
quantities are determined by the transmission probabilities of electrons in
such a way that the product is constant for a given superconductor at fixed
temperature. Our self-consistent solutions for different types of barrier
indicate that this is not the case. We suggest some forms of barrier which
could increase the Ic.R(N) product, and hence improve the frequency response of
a Josephson device.Comment: 46 pages, 21 figure
The Interaction of High-Speed Turbulence with Flames: Global Properties and Internal Flame Structure
We study the dynamics and properties of a turbulent flame, formed in the
presence of subsonic, high-speed, homogeneous, isotropic Kolmogorov-type
turbulence in an unconfined system. Direct numerical simulations are performed
with Athena-RFX, a massively parallel, fully compressible, high-order,
dimensionally unsplit, reactive-flow code. A simplified reaction-diffusion
model represents a stoichiometric H2-air mixture. The system being modeled
represents turbulent combustion with the Damkohler number Da = 0.05 and with
the turbulent velocity at the energy injection scale 30 times larger than the
laminar flame speed. The simulations show that flame interaction with
high-speed turbulence forms a steadily propagating turbulent flame with a flame
brush width approximately twice the energy injection scale and a speed four
times the laminar flame speed. A method for reconstructing the internal flame
structure is described and used to show that the turbulent flame consists of
tightly folded flamelets. The reaction zone structure of these is virtually
identical to that of the planar laminar flame, while the preheat zone is
broadened by approximately a factor of two. Consequently, the system evolution
represents turbulent combustion in the thin-reaction zone regime. The turbulent
cascade fails to penetrate the internal flame structure, and thus the action of
small-scale turbulence is suppressed throughout most of the flame. Finally, our
results suggest that for stoichiometric H2-air mixtures, any substantial flame
broadening by the action of turbulence cannot be expected in all subsonic
regimes.Comment: 30 pages, 9 figures; published in Combustion and Flam
When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks
Let be a supply graph and a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of and
demands to the edges of is said to satisfy the \emph{cut condition} if for
any cut in the graph, the total demand crossing the cut is no more than the
total capacity crossing it. The pair is called \emph{cut-sufficient} if
for any assignment of capacities and demands that satisfy the cut condition,
there is a multiflow routing the demands defined on within the network with
capacities defined on . We prove a previous conjecture, which states that
when the supply graph is series-parallel, the pair is
cut-sufficient if and only if does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of and delete edges
of and so that becomes the complete bipartite graph , with
odd, and is composed of a cycle connecting the vertices of
degree 2, and an edge connecting the two vertices of degree . We further
prove that if the instance is \emph{Eulerian} --- that is, the demands and
capacities are integers and the total of demands and capacities incident to
each vertex is even --- then the multiflow problem has an integral solution. We
provide a polynomial-time algorithm to find an integral solution in this case.
In order to prove these results, we formulate properties of tight cuts (cuts
for which the cut condition inequality is tight) in cut-sufficient pairs. We
believe these properties might be useful in extending our results to planar
graphs.Comment: An extended abstract of this paper will be published at the 44th
Symposium on Theory of Computing (STOC 2012
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