1,586 research outputs found
Distributive Lattices, Polyhedra, and Generalized Flow
A D-polyhedron is a polyhedron such that if are in then so are
their componentwise max and min. In other words, the point set of a
D-polyhedron forms a distributive lattice with the dominance order. We provide
a full characterization of the bounding hyperplanes of D-polyhedra.
Aside from being a nice combination of geometric and order theoretic
concepts, D-polyhedra are a unifying generalization of several distributive
lattices which arise from graphs. In fact every D-polyhedron corresponds to a
directed graph with arc-parameters, such that every point in the polyhedron
corresponds to a vertex potential on the graph. Alternatively, an edge-based
description of the point set can be given. The objects in this model are dual
to generalized flows, i.e., dual to flows with gains and losses.
These models can be specialized to yield some cases of distributive lattices
that have been studied previously. Particular specializations are: lattices of
flows of planar digraphs (Khuller, Naor and Klein), of -orientations of
planar graphs (Felsner), of c-orientations (Propp) and of -bonds of
digraphs (Felsner and Knauer). As an additional application we exhibit a
distributive lattice structure on generalized flow of breakeven planar
digraphs.Comment: 17 pages, 3 figure
Single Source - All Sinks Max Flows in Planar Digraphs
Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing
max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show
how to solve this problem in near-linear O(n log^3 n) time. Previously, no
better solution was known than running a single-source single-sink max flow
algorithm n-1 times, giving a total time bound of O(n^2 log n) with the
algorithm of Borradaile and Klein.
An important implication is that all-pairs max st-flow values in G can be
computed in near-quadratic time. This is close to optimal as the output size is
Theta(n^2). We give a quadratic lower bound on the number of distinct max flow
values and an Omega(n^3) lower bound for the total size of all min cut-sets.
This distinguishes the problem from the undirected case where the number of
distinct max flow values is O(n).
Previous to our result, no algorithm which could solve the all-pairs max flow
values problem faster than the time of Theta(n^2) max-flow computations for
every planar digraph was known.
This result is accompanied with a data structure that reports min cut-sets.
For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can
report the set of arcs C crossing a min st-cut in time roughly proportional to
the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201
On resilient control of dynamical flow networks
Resilience has become a key aspect in the design of contemporary
infrastructure networks. This comes as a result of ever-increasing loads,
limited physical capacity, and fast-growing levels of interconnectedness and
complexity due to the recent technological advancements. The problem has
motivated a considerable amount of research within the last few years,
particularly focused on the dynamical aspects of network flows, complementing
more classical static network flow optimization approaches. In this tutorial
paper, a class of single-commodity first-order models of dynamical flow
networks is considered. A few results recently appeared in the literature and
dealing with stability and robustness of dynamical flow networks are gathered
and originally presented in a unified framework. In particular, (differential)
stability properties of monotone dynamical flow networks are treated in some
detail, and the notion of margin of resilience is introduced as a quantitative
measure of their robustness. While emphasizing methodological aspects --
including structural properties, such as monotonicity, that enable tractability
and scalability -- over the specific applications, connections to
well-established road traffic flow models are made.Comment: accepted for publication in Annual Reviews in Control, 201
Reciprocity in Social Networks with Capacity Constraints
Directed links -- representing asymmetric social ties or interactions (e.g.,
"follower-followee") -- arise naturally in many social networks and other
complex networks, giving rise to directed graphs (or digraphs) as basic
topological models for these networks. Reciprocity, defined for a digraph as
the percentage of edges with a reciprocal edge, is a key metric that has been
used in the literature to compare different directed networks and provide
"hints" about their structural properties: for example, are reciprocal edges
generated randomly by chance or are there other processes driving their
generation? In this paper we study the problem of maximizing achievable
reciprocity for an ensemble of digraphs with the same prescribed in- and
out-degree sequences. We show that the maximum reciprocity hinges crucially on
the in- and out-degree sequences, which may be intuitively interpreted as
constraints on some "social capacities" of nodes and impose fundamental limits
on achievable reciprocity. We show that it is NP-complete to decide the
achievability of a simple upper bound on maximum reciprocity, and provide
conditions for achieving it. We demonstrate that many real networks exhibit
reciprocities surprisingly close to the upper bound, which implies that users
in these social networks are in a sense more "social" than suggested by the
empirical reciprocity alone in that they are more willing to reciprocate,
subject to their "social capacity" constraints. We find some surprising linear
relationships between empirical reciprocity and the bound. We also show that a
particular type of small network motifs that we call 3-paths are the major
source of loss in reciprocity for real networks
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