17,760 research outputs found
Combinatorial Continuous Maximal Flows
Maximum flow (and minimum cut) algorithms have had a strong impact on
computer vision. In particular, graph cuts algorithms provide a mechanism for
the discrete optimization of an energy functional which has been used in a
variety of applications such as image segmentation, stereo, image stitching and
texture synthesis. Algorithms based on the classical formulation of max-flow
defined on a graph are known to exhibit metrication artefacts in the solution.
Therefore, a recent trend has been to instead employ a spatially continuous
maximum flow (or the dual min-cut problem) in these same applications to
produce solutions with no metrication errors. However, known fast continuous
max-flow algorithms have no stopping criteria or have not been proved to
converge. In this work, we revisit the continuous max-flow problem and show
that the analogous discrete formulation is different from the classical
max-flow problem. We then apply an appropriate combinatorial optimization
technique to this combinatorial continuous max-flow CCMF problem to find a
null-divergence solution that exhibits no metrication artefacts and may be
solved exactly by a fast, efficient algorithm with provable convergence.
Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the
fact, already proved by Nozawa in the continuous setting, that the max-flow and
the total variation problems are not always equivalent.Comment: 26 page
Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness
Hedonic pricing with quasi-linear preferences is shown to be equivalent to stable matching with transferable utilities and a participation constraint, and to an optimal transportation (Monge-Kantorovich) linear programming problem. Optimal assignments in the latter correspond to stable matchings, and to hedonic equilibria. These assignments are shown to exist in great generality; their marginal indirect payoffs with respect to agent type are shown to be unique whenever direct payoffs vary smoothly with type. Under a generalized Spence-Mirrlees condition (also known as a twist condition) the assignments are shown to be unique and to be pure, meaning the matching is one-to-one outside a negligible set. For smooth problems set on compact, connected type spaces such as the circle, there is a topological obstruction to purity, but we give a weaker condition still guaranteeing uniqueness of the stable match
Cross-layer Congestion Control, Routing and Scheduling Design in Ad Hoc Wireless Networks
This paper considers jointly optimal design of crosslayer congestion control, routing and scheduling for ad hoc
wireless networks. We first formulate the rate constraint and scheduling constraint using multicommodity flow variables, and formulate resource allocation in networks with fixed wireless channels (or single-rate wireless devices that can mask channel variations) as a utility maximization problem with these constraints.
By dual decomposition, the resource allocation problem
naturally decomposes into three subproblems: congestion control,
routing and scheduling that interact through congestion price.
The global convergence property of this algorithm is proved. We
next extend the dual algorithm to handle networks with timevarying
channels and adaptive multi-rate devices. The stability
of the resulting system is established, and its performance is
characterized with respect to an ideal reference system which
has the best feasible rate region at link layer.
We then generalize the aforementioned results to a general
model of queueing network served by a set of interdependent
parallel servers with time-varying service capabilities, which
models many design problems in communication networks. We
show that for a general convex optimization problem where a
subset of variables lie in a polytope and the rest in a convex set,
the dual-based algorithm remains stable and optimal when the
constraint set is modulated by an irreducible finite-state Markov
chain. This paper thus presents a step toward a systematic way
to carry out cross-layer design in the framework of “layering as
optimization decomposition” for time-varying channel models
A spectral lower bound for the divisorial gonality of metric graphs
Let be a compact metric graph, and denote by the Laplace
operator on with the first non-trivial eigenvalue . We
prove the following Yang-Li-Yau type inequality on divisorial gonality
of . There is a universal constant such that
where the
volume is the total length of the edges in ,
is the minimum length of all the geodesic paths
between points of of valence different from two, and is the
largest valence of points of . Along the way, we also establish
discrete versions of the above inequality concerning finite simple graph models
of and their spectral gaps.Comment: 22 pages, added new recent references, minor revisio
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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