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Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
Random curves on surfaces induced from the Laplacian determinant
We define natural probability measures on cycle-rooted spanning forests
(CRSFs) on graphs embedded on a surface with a Riemannian metric. These
measures arise from the Laplacian determinant and depend on the choice of a
unitary connection on the tangent bundle to the surface.
We show that, for a sequence of graphs conformally approximating the
surface, the measures on CRSFs of converge and give a limiting
probability measure on finite multicurves (finite collections of pairwise
disjoint simple closed curves) on the surface, independent of the approximating
sequence.
Wilson's algorithm for generating spanning trees on a graph generalizes to a
cycle-popping algorithm for generating CRSFs for a general family of weights on
the cycles. We use this to sample the above measures. The sampling algorithm,
which relates these measures to the loop-erased random walk, is also used to
prove tightness of the sequence of measures, a key step in the proof of their
convergence.
We set the framework for the study of these probability measures and their
scaling limits and state some of their properties
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
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