16,947 research outputs found
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Continuous Multiclass Labeling Approaches and Algorithms
We study convex relaxations of the image labeling problem on a continuous
domain with regularizers based on metric interaction potentials. The generic
framework ensures existence of minimizers and covers a wide range of
relaxations of the originally combinatorial problem. We focus on two specific
relaxations that differ in flexibility and simplicity -- one can be used to
tightly relax any metric interaction potential, while the other one only covers
Euclidean metrics but requires less computational effort. For solving the
nonsmooth discretized problem, we propose a globally convergent
Douglas-Rachford scheme, and show that a sequence of dual iterates can be
recovered in order to provide a posteriori optimality bounds. In a quantitative
comparison to two other first-order methods, the approach shows competitive
performance on synthetical and real-world images. By combining the method with
an improved binarization technique for nonstandard potentials, we were able to
routinely recover discrete solutions within 1%--5% of the global optimum for
the combinatorial image labeling problem
Block Stability for MAP Inference
To understand the empirical success of approximate MAP inference, recent work
(Lang et al., 2018) has shown that some popular approximation algorithms
perform very well when the input instance is stable. The simplest stability
condition assumes that the MAP solution does not change at all when some of the
pairwise potentials are (adversarially) perturbed. Unfortunately, this strong
condition does not seem to be satisfied in practice. In this paper, we
introduce a significantly more relaxed condition that only requires blocks
(portions) of an input instance to be stable. Under this block stability
condition, we prove that the pairwise LP relaxation is persistent on the stable
blocks. We complement our theoretical results with an empirical evaluation of
real-world MAP inference instances from computer vision. We design an algorithm
to find stable blocks, and find that these real instances have large stable
regions. Our work gives a theoretical explanation for the widespread empirical
phenomenon of persistency for this LP relaxation
Fast Construction of Nets in Low Dimensional Metrics, and Their Applications
We present a near linear time algorithm for constructing hierarchical nets in
finite metric spaces with constant doubling dimension. This data-structure is
then applied to obtain improved algorithms for the following problems:
Approximate nearest neighbor search, well-separated pair decomposition, compact
representation scheme, doubling measure, and computation of the (approximate)
Lipschitz constant of a function. In all cases, the running (preprocessing)
time is near-linear and the space being used is linear.Comment: 41 pages. Extensive clean-up of minor English error
Route Planning in Transportation Networks
We survey recent advances in algorithms for route planning in transportation
networks. For road networks, we show that one can compute driving directions in
milliseconds or less even at continental scale. A variety of techniques provide
different trade-offs between preprocessing effort, space requirements, and
query time. Some algorithms can answer queries in a fraction of a microsecond,
while others can deal efficiently with real-time traffic. Journey planning on
public transportation systems, although conceptually similar, is a
significantly harder problem due to its inherent time-dependent and
multicriteria nature. Although exact algorithms are fast enough for interactive
queries on metropolitan transit systems, dealing with continent-sized instances
requires simplifications or heavy preprocessing. The multimodal route planning
problem, which seeks journeys combining schedule-based transportation (buses,
trains) with unrestricted modes (walking, driving), is even harder, relying on
approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4,
previously published by Microsoft Research. This work was mostly done while
the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at
Microsoft Research Silicon Valle
Sublinear Distance Labeling
A distance labeling scheme labels the nodes of a graph with binary
strings such that, given the labels of any two nodes, one can determine the
distance in the graph between the two nodes by looking only at the labels. A
-preserving distance labeling scheme only returns precise distances between
pairs of nodes that are at distance at least from each other. In this paper
we consider distance labeling schemes for the classical case of unweighted
graphs with both directed and undirected edges.
We present a bit -preserving distance labeling
scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete
Math. 2005]. We also give an almost matching lower bound of
. With our -preserving distance labeling scheme as a
building block, we additionally achieve the following results:
1. We present the first distance labeling scheme of size for sparse
graphs (and hence bounded degree graphs). This addresses an open problem by
Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from
distance labeling in general graphs which require bits, Moon [Proc.
of Glasgow Math. Association 1965].
2. For approximate -additive labeling schemes, that return distances
within an additive error of we show a scheme of size for .
This improves on the current best bound of by
Alstrup et. al. [SODA 2016] for sub-polynomial , and is a generalization of
a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for
.Comment: A preliminary version of this paper appeared at ESA'1
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