3 research outputs found
Geometric Dominating Set and Set Cover via Local Search
In this paper, we study two classic optimization problems: minimum geometric
dominating set and set cover. Both the problems have been studied for different
types of objects for a long time. These problems become APX-hard when the
objects are axis-parallel rectangles, ellipses, -fat objects of
constant description complexity, and convex polygons. On the other hand, PTAS
(polynomial time approximation scheme) is known for them when the objects are
disks or unit squares. Surprisingly, PTAS was unknown even for arbitrary
squares. For homothetic set of convex objects, an approximation
algorithm is known for dominating set problem, where is the number of
corners in a convex object. On the other hand, QPTAS (quasi polynomial time
approximation scheme) is known very recently for the covering problem when the
objects are pseudodisks. For both problems obtaining a PTAS remains open for a
large class of objects.
For the dominating set problems, we prove that the popular local search
algorithm leads to an approximation when objects are
homothetic set of convex objects (which includes arbitrary squares, -regular
polygons, translated and scaled copies of a convex set etc.) in
time. On the other hand, the same technique leads to a
PTAS for geometric covering problem when the objects are convex pseudodisks
(which includes disks, unit height rectangles, homothetic convex objects etc.).
As a consequence, we obtain an easy to implement approximation algorithm for
both problems for a large class of objects, significantly improving the best
known approximation guarantees.Comment: 25 pages, 4 figure
Algorithms for Intersection Graphs of Multiple Intervals and Pseudo Disks
Intersection graphs of planar geometric objects such as intervals, disks,
rectangles and pseudo-disks are well studied. Motivated by various
applications, Butman et al. in SODA 2007 considered algorithmic questions in
intersection graphs of -intervals. A -interval is a union of at most
distinct intervals (here is a parameter) -- these graphs are referred to as
Multiple-Interval Graphs. Subsequent work by Kammer et al. in Approx 2010 also
considered -disks and other geometric shapes. In this paper we revisit some
of these algorithmic questions via more recent developments in computational
geometry. For the minimum weight dominating set problem, we give a simple approximation for Multiple-Interval Graphs, improving on the
previously known bound of . We also show that it is NP-hard to obtain an
-approximation in this case. In fact, our results hold for the
intersection graph of a set of t-pseudo-disks which is a much larger class. We
obtain an -approximation for the maximum weight independent set
in the intersection graph of -pseudo-disks. Our results are based on simple
reductions to existing algorithms by appropriately bounding the union
complexity of the objects under consideration
The Robust Minimal Controllability Problem
In this paper, we address two minimal controllability problems, where the
goal is to determine a minimal subset of state variables in a linear
time-invariant system to be actuated to ensure controllability under additional
constraints. First, we study the problem of characterizing the sparsest input
matrices that assure controllability when the autonomous dynamics' matrix is
simple. Secondly, we build upon these results to describe the solutions to the
robust minimal controllability problem, where the goal is to determine the
sparsest input matrix ensuring controllability when specified number of inputs
fail. Both problems are NP-hard, but under the assumption that the dynamics'
matrix is simple, we show that it is possible to reduce these two problems to
set multi-covering problems. Consequently, these problems share the same
computational complexity, i.e., they are NP-complete, but polynomial algorithms
to approximate the solutions of a set multi-covering problem can be leveraged
to obtain close-to-optimal solutions to either of the minimal controllability
problems