4,834 research outputs found
Symmetric Submodular Function Minimization Under Hereditary Family Constraints
We present an efficient algorithm to find non-empty minimizers of a symmetric
submodular function over any family of sets closed under inclusion. This for
example includes families defined by a cardinality constraint, a knapsack
constraint, a matroid independence constraint, or any combination of such
constraints. Our algorithm make oracle calls to the submodular
function where is the cardinality of the ground set. In contrast, the
problem of minimizing a general submodular function under a cardinality
constraint is known to be inapproximable within (Svitkina
and Fleischer [2008]).
The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to
find all nontrivial inclusionwise minimal minimizers of a symmetric submodular
function over a set of cardinality using oracle calls. Their
procedure in turn is based on Queyranne's algorithm [1998] to minimize a
symmetric submodularComment: 13 pages, Submitted to SODA 201
Planar Object Tracking in the Wild: A Benchmark
Planar object tracking is an actively studied problem in vision-based robotic
applications. While several benchmarks have been constructed for evaluating
state-of-the-art algorithms, there is a lack of video sequences captured in the
wild rather than in constrained laboratory environment. In this paper, we
present a carefully designed planar object tracking benchmark containing 210
videos of 30 planar objects sampled in the natural environment. In particular,
for each object, we shoot seven videos involving various challenging factors,
namely scale change, rotation, perspective distortion, motion blur, occlusion,
out-of-view, and unconstrained. The ground truth is carefully annotated
semi-manually to ensure the quality. Moreover, eleven state-of-the-art
algorithms are evaluated on the benchmark using two evaluation metrics, with
detailed analysis provided for the evaluation results. We expect the proposed
benchmark to benefit future studies on planar object tracking.Comment: Accepted by ICRA 201
Submodular Minimization Under Congruency Constraints
Submodular function minimization (SFM) is a fundamental and efficiently
solvable problem class in combinatorial optimization with a multitude of
applications in various fields. Surprisingly, there is only very little known
about constraint types under which SFM remains efficiently solvable. The
arguably most relevant non-trivial constraint class for which polynomial SFM
algorithms are known are parity constraints, i.e., optimizing only over sets of
odd (or even) cardinality. Parity constraints capture classical combinatorial
optimization problems like the odd-cut problem, and they are a key tool in a
recent technique to efficiently solve integer programs with a constraint matrix
whose subdeterminants are bounded by two in absolute value.
We show that efficient SFM is possible even for a significantly larger class
than parity constraints, by introducing a new approach that combines techniques
from Combinatorial Optimization, Combinatorics, and Number Theory. In
particular, we can show that efficient SFM is possible over all sets (of any
given lattice) of cardinality r mod m, as long as m is a constant prime power.
This covers generalizations of the odd-cut problem with open complexity status,
and with relevance in the context of integer programming with higher
subdeterminants. To obtain our results, we establish a connection between the
correctness of a natural algorithm, and the inexistence of set systems with
specific combinatorial properties. We introduce a general technique to disprove
the existence of such set systems, which allows for obtaining extensions of our
results beyond the above-mentioned setting. These extensions settle two open
questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of
computing the girth and cogirth of certain types of binary matroids
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