14 research outputs found
An average case analysis of the minimum spanning tree heuristic for the range assignment problem
We present an average case analysis of the minimum spanning tree heuristic for the range assignment problem on a graph with power weighted edges. It is well-known that the worst-case approximation ratio of this heuristic is 2. Our analysis yields the following results: (1) In the one dimensional case (), where the weights of the edges are 1 with probability and 0 otherwise, the average-case approximation ratio is bounded from above by . (2) When and the distance between neighboring vertices is drawn from a uniform -distribution, the average approximation ratio is bounded from above by where denotes the distance power radient. (3) In Euclidean 2-dimensional space, with distance power gradient , the average performance ratio is bounded from above by
The Traveling Salesman Problem Under Squared Euclidean Distances
Let be a set of points in , and let be a
real number. We define the distance between two points as
, where denotes the standard Euclidean distance between
and . We denote the traveling salesman problem under this distance
function by TSP(). We design a 5-approximation algorithm for TSP(2,2)
and generalize this result to obtain an approximation factor of
for and all .
We also study the variant Rev-TSP of the problem where the traveling salesman
is allowed to revisit points. We present a polynomial-time approximation scheme
for Rev-TSP with , and we show that Rev-TSP is APX-hard if and . The APX-hardness proof carries
over to TSP for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change
Compact and Extended Formulations for Range Assignment Problems
We devise two new integer programming models for range assignment problems arising in wireless network design. Building on an arbitrary set of feasible network topologies, e.g., all spanning trees, we explicitly model the power consumption at a given node as a weighted maximum over edge variables. We show that the standard ILP model is an extended formulation of the new models. For all models, we derive complete polyhedral descriptions in the unconstrained case where all topologies are allowed. These results give rise to tight relaxations even in the constrained case. We can show experimentally that the compact formulations compare favorably to the standard approach
Compact and Extended Formulations for Range Assignment Problems
We devise two new integer programming models for range assignment problems arising in wireless network design. Building on an arbitrary set of feasible network topologies, e.g., all spanning trees, we explicitly model the power consumption at a given node as a weighted maximum over edge variables. We show that the standard ILP model is an extended formulation of the new models. For all models, we derive complete polyhedral descriptions in the unconstrained case where all topologies are allowed. These results give rise to tight relaxations even in the constrained case. We can show experimentally that the compact formulations compare favorably to the standard approach
The Online Broadcast Range-Assignment Problem
Let be a set of points in , modeling
devices in a wireless network. A range assignment assigns a range to
each point , thus inducing a directed communication graph in
which there is a directed edge iff , where denotes the distance between and
. The range-assignment problem is to assign the transmission ranges such
that has a certain desirable property, while minimizing the cost of the
assignment; here the cost is given by , for
some constant called the distance-power gradient.
We introduce the online version of the range-assignment problem, where the
points arrive one by one, and the range assignment has to be updated at
each arrival. Following the standard in online algorithms, resources given out
cannot be taken away -- in our case this means that the transmission ranges
will never decrease. The property we want to maintain is that has a
broadcast tree rooted at the first point . Our results include the
following.
- For , a 1-competitive algorithm does not exist. In particular, for
any online algorithm has competitive ratio at least 1.57.
- For and , we analyze two natural strategies: Upon the arrival of
a new point , Nearest-Neighbor increases the range of the nearest point to
cover and Cheapest Increase increases the range of the point for which
the resulting cost increase to be able to reach is minimal.
- We generalize the problem to arbitrary metric spaces, where we present an
-competitive algorithm.Comment: Preliminary version in ISAAC 202
The Homogeneous Broadcast Problem in Narrow and Wide Strips
Let be a set of nodes in a wireless network, where each node is modeled
as a point in the plane, and let be a given source node. Each node
can transmit information to all other nodes within unit distance, provided
is activated. The (homogeneous) broadcast problem is to activate a minimum
number of nodes such that in the resulting directed communication graph, the
source can reach any other node. We study the complexity of the regular and
the hop-bounded version of the problem (in the latter, must be able to
reach every node within a specified number of hops), with the restriction that
all points lie inside a strip of width . We almost completely characterize
the complexity of both the regular and the hop-bounded versions as a function
of the strip width .Comment: 50 pages, WADS 2017 submissio
Connecting a Set of Circles with Minimum Sum of Radii
Abstract. We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NP-hard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomialtime algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomial-time approximation schemes, and conclude with experimental results and open problems