79,867 research outputs found
Smoothed Analysis of the Successive Shortest Path Algorithm
The minimum-cost flow problem is a classic problem in combinatorial
optimization with various applications. Several pseudo-polynomial, polynomial,
and strongly polynomial algorithms have been developed in the past decades, and
it seems that both the problem and the algorithms are well understood. However,
some of the algorithms' running times observed in empirical studies contrast
the running times obtained by worst-case analysis not only in the order of
magnitude but also in the ranking when compared to each other. For example, the
Successive Shortest Path (SSP) algorithm, which has an exponential worst-case
running time, seems to outperform the strongly polynomial Minimum-Mean Cycle
Canceling algorithm.
To explain this discrepancy, we study the SSP algorithm in the framework of
smoothed analysis and establish a bound of for the number of
iterations, which implies a smoothed running time of ,
where and denote the number of nodes and edges, respectively, and
is a measure for the amount of random noise. This shows that worst-case
instances for the SSP algorithm are not robust and unlikely to be encountered
in practice. Furthermore, we prove a smoothed lower bound of for the number of iterations of the SSP algorithm, showing
that the upper bound cannot be improved for .Comment: A preliminary version has been presented at SODA 201
Network Flow Algorithms for Structured Sparsity
We consider a class of learning problems that involve a structured
sparsity-inducing norm defined as the sum of -norms over groups of
variables. Whereas a lot of effort has been put in developing fast optimization
methods when the groups are disjoint or embedded in a specific hierarchical
structure, we address here the case of general overlapping groups. To this end,
we show that the corresponding optimization problem is related to network flow
optimization. More precisely, the proximal problem associated with the norm we
consider is dual to a quadratic min-cost flow problem. We propose an efficient
procedure which computes its solution exactly in polynomial time. Our algorithm
scales up to millions of variables, and opens up a whole new range of
applications for structured sparse models. We present several experiments on
image and video data, demonstrating the applicability and scalability of our
approach for various problems.Comment: accepted for publication in Adv. Neural Information Processing
Systems, 201
Belief Propagation Min-Sum Algorithm for Generalized Min-Cost Network Flow
Belief Propagation algorithms are instruments used broadly to solve graphical
model optimization and statistical inference problems. In the general case of a
loopy Graphical Model, Belief Propagation is a heuristic which is quite
successful in practice, even though its empirical success, typically, lacks
theoretical guarantees. This paper extends the short list of special cases
where correctness and/or convergence of a Belief Propagation algorithm is
proven. We generalize formulation of Min-Sum Network Flow problem by relaxing
the flow conservation (balance) constraints and then proving that the Belief
Propagation algorithm converges to the exact result
Combinatorial persistency criteria for multicut and max-cut
In combinatorial optimization, partial variable assignments are called
persistent if they agree with some optimal solution. We propose persistency
criteria for the multicut and max-cut problem as well as fast combinatorial
routines to verify them. The criteria that we derive are based on mappings that
improve feasible multicuts, respectively cuts. Our elementary criteria can be
checked enumeratively. The more advanced ones rely on fast algorithms for upper
and lower bounds for the respective cut problems and max-flow techniques for
auxiliary min-cut problems. Our methods can be used as a preprocessing
technique for reducing problem sizes or for computing partial optimality
guarantees for solutions output by heuristic solvers. We show the efficacy of
our methods on instances of both problems from computer vision, biomedical
image analysis and statistical physics
NeuRoute: Predictive Dynamic Routing for Software-Defined Networks
This paper introduces NeuRoute, a dynamic routing framework for Software
Defined Networks (SDN) entirely based on machine learning, specifically, Neural
Networks. Current SDN/OpenFlow controllers use a default routing based on
Dijkstra algorithm for shortest paths, and provide APIs to develop custom
routing applications. NeuRoute is a controller-agnostic dynamic routing
framework that (i) predicts traffic matrix in real time, (ii) uses a neural
network to learn traffic characteristics and (iii) generates forwarding rules
accordingly to optimize the network throughput. NeuRoute achieves the same
results as the most efficient dynamic routing heuristic but in much less
execution time.Comment: Accepted for CNSM 201
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