2,006 research outputs found
Spider covers for prize-collecting network activation problem
In the network activation problem, each edge in a graph is associated with an
activation function, that decides whether the edge is activated from
node-weights assigned to its end-nodes. The feasible solutions of the problem
are the node-weights such that the activated edges form graphs of required
connectivity, and the objective is to find a feasible solution minimizing its
total weight. In this paper, we consider a prize-collecting version of the
network activation problem, and present first non- trivial approximation
algorithms. Our algorithms are based on a new LP relaxation of the problem.
They round optimal solutions for the relaxation by repeatedly computing
node-weights activating subgraphs called spiders, which are known to be useful
for approximating the network activation problem
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A parallel genetic algorithm for the Steiner Problem in Networks
This paper presents a parallel genetic algorithm to the
Steiner Problem in Networks. Several previous papers
have proposed the adoption of GAs and others
metaheuristics to solve the SPN demonstrating the
validity of their approaches. This work differs from them
for two main reasons: the dimension and the
characteristics of the networks adopted in the experiments
and the aim from which it has been originated. The reason
that aimed this work was namely to build a comparison
term for validating deterministic and computationally
inexpensive algorithms which can be used in practical
engineering applications, such as the multicast
transmission in the Internet. On the other hand, the large
dimensions of our sample networks require the adoption
of a parallel implementation of the Steiner GA, which is
able to deal with such large problem instances
Fast Routing Table Construction Using Small Messages
We describe a distributed randomized algorithm computing approximate
distances and routes that approximate shortest paths. Let n denote the number
of nodes in the graph, and let HD denote the hop diameter of the graph, i.e.,
the diameter of the graph when all edges are considered to have unit weight.
Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD)
communication rounds using messages of O(log n) bits and guarantees a stretch
of O(eps^(-1) log eps^(-1)) with high probability. This is the first
distributed algorithm approximating weighted shortest paths that uses small
messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time
complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the
small-messages model that hold for stateless routing (where routing decisions
do not depend on the traversed path) as well as approximation of the weigthed
diameter. Our scheme replaces the original identifiers of the nodes by labels
of size O(log eps^(-1) log n). We show that no algorithm that keeps the
original identifiers and runs for weak-o(n) rounds can achieve a
polylogarithmic approximation ratio.
Variations of our techniques yield a number of fast distributed approximation
algorithms solving related problems using small messages. Specifically, we
present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0
< eps <= 1/2, and solve, with high probability, the following problems:
- O(eps^(-1))-approximation for the Generalized Steiner Forest (the running
time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the
number of terminals);
- O(eps^(-2))-approximation of weighted distances, using node labels of size
O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node;
- O(eps^(-1))-approximation of the weighted diameter;
- O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1
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
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