1,091,571 research outputs found
Covering problems in edge- and node-weighted graphs
This paper discusses the graph covering problem in which a set of edges in an
edge- and node-weighted graph is chosen to satisfy some covering constraints
while minimizing the sum of the weights. In this problem, because of the large
integrality gap of a natural linear programming (LP) relaxation, LP rounding
algorithms based on the relaxation yield poor performance. Here we propose a
stronger LP relaxation for the graph covering problem. The proposed relaxation
is applied to designing primal-dual algorithms for two fundamental graph
covering problems: the prize-collecting edge dominating set problem and the
multicut problem in trees. Our algorithms are an exact polynomial-time
algorithm for the former problem, and a 2-approximation algorithm for the
latter problem, respectively. These results match the currently known best
results for purely edge-weighted graphs.Comment: To appear in SWAT 201
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
Restricted Strip Covering and the Sensor Cover Problem
Given a set of objects with durations (jobs) that cover a base region, can we
schedule the jobs to maximize the duration the original region remains covered?
We call this problem the sensor cover problem. This problem arises in the
context of covering a region with sensors. For example, suppose you wish to
monitor activity along a fence by sensors placed at various fixed locations.
Each sensor has a range and limited battery life. The problem is to schedule
when to turn on the sensors so that the fence is fully monitored for as long as
possible. This one dimensional problem involves intervals on the real line.
Associating a duration to each yields a set of rectangles in space and time,
each specified by a pair of fixed horizontal endpoints and a height. The
objective is to assign a position to each rectangle to maximize the height at
which the spanning interval is fully covered. We call this one dimensional
problem restricted strip covering. If we replace the covering constraint by a
packing constraint, the problem is identical to dynamic storage allocation, a
scheduling problem that is a restricted case of the strip packing problem. We
show that the restricted strip covering problem is NP-hard and present an O(log
log n)-approximation algorithm. We present better approximations or exact
algorithms for some special cases. For the uniform-duration case of restricted
strip covering we give a polynomial-time, exact algorithm but prove that the
uniform-duration case for higher-dimensional regions is NP-hard. Finally, we
consider regions that are arbitrary sets, and we present an O(log
n)-approximation algorithm.Comment: 14 pages, 6 figure
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
Consider the following problem: given a set system (U,I) and an edge-weighted
graph G = (U, E) on the same universe U, find the set A in I such that the
Steiner tree cost with terminals A is as large as possible: "which set in I is
the most difficult to connect up?" This is an example of a max-min problem:
find the set A in I such that the value of some minimization (covering) problem
is as large as possible.
In this paper, we show that for certain covering problems which admit good
deterministic online algorithms, we can give good algorithms for max-min
optimization when the set system I is given by a p-system or q-knapsacks or
both. This result is similar to results for constrained maximization of
submodular functions. Although many natural covering problems are not even
approximately submodular, we show that one can use properties of the online
algorithm as a surrogate for submodularity.
Moreover, we give stronger connections between max-min optimization and
two-stage robust optimization, and hence give improved algorithms for robust
versions of various covering problems, for cases where the uncertainty sets are
given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and
http://arxiv.org/abs/0912.1045 appeared in ICALP 201
A note on the greedy approximation algorithm for the unweighted set covering problem
Bibliography: leaf 9.Abhay K. Parekh
On some covering problems in geometry
We present a method to obtain upper bounds on covering numbers. As
applications of this method, we reprove and generalize results of Rogers on
economically covering Euclidean -space with translates of a convex body, or
more generally, any measurable set. We obtain a bound for the density of
covering the -sphere by rotated copies of a spherically convex set (or, any
measurable set). Using the same method, we sharpen an estimate by
Artstein--Avidan and Slomka on covering a bounded set by translates of another.
The main novelty of our method is that it is not probabilistic. The key idea,
which makes our proofs rather simple and uniform through different settings, is
an algorithmic result of Lov\'asz and Stein.Comment: 9 pages. IMPORTANT CHANGE: In previous versions of the paper, the
illumination problem was also considered, and I presented a construction of a
body close to the Euclidean ball with high illumination number. Now, I
removed this part from this manuscript and made it a separate paper, 'A Spiky
Ball'. It can be found at http://arxiv.org/abs/1510.0078
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