14,082 research outputs found
Polynomial-Time Space-Optimal Silent Self-Stabilizing Minimum-Degree Spanning Tree Construction
Motivated by applications to sensor networks, as well as to many other areas,
this paper studies the construction of minimum-degree spanning trees. We
consider the classical node-register state model, with a weakly fair scheduler,
and we present a space-optimal \emph{silent} self-stabilizing construction of
minimum-degree spanning trees in this model. Computing a spanning tree with
minimum degree is NP-hard. Therefore, we actually focus on constructing a
spanning tree whose degree is within one from the optimal. Our algorithm uses
registers on bits, converges in a polynomial number of rounds, and
performs polynomial-time computation at each node. Specifically, the algorithm
constructs and stabilizes on a special class of spanning trees, with degree at
most . Indeed, we prove that, unless NP coNP, there are no
proof-labeling schemes involving polynomial-time computation at each node for
the whole family of spanning trees with degree at most . Up to our
knowledge, this is the first example of the design of a compact silent
self-stabilizing algorithm constructing, and stabilizing on a subset of optimal
solutions to a natural problem for which there are no time-efficient
proof-labeling schemes. On our way to design our algorithm, we establish a set
of independent results that may have interest on their own. In particular, we
describe a new space-optimal silent self-stabilizing spanning tree
construction, stabilizing on \emph{any} spanning tree, in rounds, and
using just \emph{one} additional bit compared to the size of the labels used to
certify trees. We also design a silent loop-free self-stabilizing algorithm for
transforming a tree into another tree. Last but not least, we provide a silent
self-stabilizing algorithm for computing and certifying the labels of a
NCA-labeling scheme
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
Mixed Map Labeling
Point feature map labeling is a geometric problem, in which a set of input
points must be labeled with a set of disjoint rectangles (the bounding boxes of
the label texts). Typically, labeling models either use internal labels, which
must touch their feature point, or external (boundary) labels, which are placed
on one of the four sides of the input points' bounding box and which are
connected to their feature points by crossing-free leader lines. In this paper
we study polynomial-time algorithms for maximizing the number of internal
labels in a mixed labeling model that combines internal and external labels.
The model requires that all leaders are parallel to a given orientation , whose value influences the geometric properties and hence the
running times of our algorithms.Comment: Full version for the paper accepted at CIAC 201
Trajectory-Based Dynamic Map Labeling
In this paper we introduce trajectory-based labeling, a new variant of
dynamic map labeling, where a movement trajectory for the map viewport is
given. We define a general labeling model and study the active range
maximization problem in this model. The problem is NP-complete and W[1]-hard.
In the restricted, yet practically relevant case that no more than k labels can
be active at any time, we give polynomial-time algorithms. For the general case
we present a practical ILP formulation with an experimental evaluation as well
as approximation algorithms.Comment: 19 pages, 7 figures, extended version of a paper to appear at ISAAC
201
Narrow sieves for parameterized paths and packings
We present randomized algorithms for some well-studied, hard combinatorial
problems: the k-path problem, the p-packing of q-sets problem, and the
q-dimensional p-matching problem. Our algorithms solve these problems with high
probability in time exponential only in the parameter (k, p, q) and using
polynomial space; the constant bases of the exponentials are significantly
smaller than in previous works. For example, for the k-path problem the
improvement is from 2 to 1.66. We also show how to detect if a d-regular graph
admits an edge coloring with colors in time within a polynomial factor of
O(2^{(d-1)n/2}).
Our techniques build upon and generalize some recently published ideas by I.
Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj\"orklund (STACS 2010,
FOCS 2010)
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