6,868 research outputs found
Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees
Given a graph , we wish to compute a spanning tree whose maximum
vertex degree, i.e. tree degree, is as small as possible. Computing the exact
optimal solution is known to be NP-hard, since it generalizes the Hamiltonian
path problem. For the approximation version of this problem, a
time algorithm that computes a spanning tree of degree at most is
previously known [F\"urer \& Raghavachari 1994]; here denotes the
minimum tree degree of all the spanning trees. In this paper we give the first
near-linear time approximation algorithm for this problem. Specifically
speaking, we propose an time algorithm that
computes a spanning tree with tree degree for any constant .
Thus, when , we can achieve approximate solutions with
constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page
The cavity approach for Steiner trees packing problems
The Belief Propagation approximation, or cavity method, has been recently
applied to several combinatorial optimization problems in its zero-temperature
implementation, the max-sum algorithm. In particular, recent developments to
solve the edge-disjoint paths problem and the prize-collecting Steiner tree
problem on graphs have shown remarkable results for several classes of graphs
and for benchmark instances. Here we propose a generalization of these
techniques for two variants of the Steiner trees packing problem where multiple
"interacting" trees have to be sought within a given graph. Depending on the
interaction among trees we distinguish the vertex-disjoint Steiner trees
problem, where trees cannot share nodes, from the edge-disjoint Steiner trees
problem, where edges cannot be shared by trees but nodes can be members of
multiple trees. Several practical problems of huge interest in network design
can be mapped into these two variants, for instance, the physical design of
Very Large Scale Integration (VLSI) chips. The formalism described here relies
on two components edge-variables that allows us to formulate a massage-passing
algorithm for the V-DStP and two algorithms for the E-DStP differing in the
scaling of the computational time with respect to some relevant parameters. We
will show that one of the two formalisms used for the edge-disjoint variant
allow us to map the max-sum update equations into a weighted maximum matching
problem over proper bipartite graphs. We developed a heuristic procedure based
on the max-sum equations that shows excellent performance in synthetic networks
(in particular outperforming standard multi-step greedy procedures by large
margins) and on large benchmark instances of VLSI for which the optimal
solution is known, on which the algorithm found the optimum in two cases and
the gap to optimality was never larger than 4 %
Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs
Let be a strongly connected directed graph. We consider the following
three problems, where we wish to compute the smallest strongly connected
spanning subgraph of that maintains respectively: the -edge-connected
blocks of (\textsf{2EC-B}); the -edge-connected components of
(\textsf{2EC-C}); both the -edge-connected blocks and the -edge-connected
components of (\textsf{2EC-B-C}). All three problems are NP-hard, and thus
we are interested in efficient approximation algorithms. For \textsf{2EC-C} we
can obtain a -approximation by combining previously known results. For
\textsf{2EC-B} and \textsf{2EC-B-C}, we present new -approximation
algorithms that run in linear time. We also propose various heuristics to
improve the size of the computed subgraphs in practice, and conduct a thorough
experimental study to assess their merits in practical scenarios
- …