153,408 research outputs found
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 %
Landmark Guided Probabilistic Roadmap Queries
A landmark based heuristic is investigated for reducing query phase run-time
of the probabilistic roadmap (\PRM) motion planning method. The heuristic is
generated by storing minimum spanning trees from a small number of vertices
within the \PRM graph and using these trees to approximate the cost of a
shortest path between any two vertices of the graph. The intermediate step of
preprocessing the graph increases the time and memory requirements of the
classical motion planning technique in exchange for speeding up individual
queries making the method advantageous in multi-query applications. This paper
investigates these trade-offs on \PRM graphs constructed in randomized
environments as well as a practical manipulator simulation.We conclude that the
method is preferable to Dijkstra's algorithm or the algorithm with
conventional heuristics in multi-query applications.Comment: 7 Page
Random tree growth by vertex splitting
We study a model of growing planar tree graphs where in each time step we
separate the tree into two components by splitting a vertex and then connect
the two pieces by inserting a new link between the daughter vertices. This
model generalises the preferential attachment model and Ford's -model
for phylogenetic trees. We develop a mean field theory for the vertex degree
distribution, prove that the mean field theory is exact in some special cases
and check that it agrees with numerical simulations in general. We calculate
various correlation functions and show that the intrinsic Hausdorff dimension
can vary from one to infinity, depending on the parameters of the model.Comment: 47 page
Coloring of two-step graphs: open packing partitioning of graphs
The two-step graphs are revisited by studying their chromatic numbers in this
paper. We observe that the problem of coloring of two-step graphs is equivalent
to the problem of vertex partitioning of graphs into open packing sets. With
this remark in mind, it can be considered as the open version of the well-known
-distance coloring problem as well as the dual version of total domatic
problem.
The minimum for which the two-step graph of a graph
admits a proper coloring assigning colors to the vertices is called the
open packing partition number of , that is,
p_{o}(G)=\chi\big{(}\mathcal{N}(G)\big{)}. We give some sharp lower and upper
bounds on this parameter as well as its exact value when dealing with some
families of graphs like trees. Relations between and some well-know
graph parameters have been investigated in this paper. We study this vertex
partitioning in the Cartesian, direct and lexicographic products of graphs. In
particular, we give an exact formula in the case of lexicographic product of
any two graphs. The NP-hardness of the problem of computing this parameter is
derived from the mentioned formula. Graphs for which equals the
clique number of are also investigated
The 3-path-step operator on trees and unicyclic graphs
summary:E. Prisner in his book Graph Dynamics defines the -path-step operator on the class of finite graphs. The -path-step operator (for a positive integer ) is the operator which to every finite graph assigns the graph which has the same vertex set as and in which two vertices are adjacent if and only if there exists a path of length in connecting them. In the paper the trees and the unicyclic graphs fixed in the operator are studied
Multiple message broadcasting and gossiping in the dynamically orientable graphs
This research investigates the problems of gossiping and multiple message broadcasting in dynamically orientable graphs of different network topologies. These are new problems never attempted before. Dynamically orientable graphs and six different network topologies are considered: paths, cycles, stars, binary trees, complete trees and two-dimensional grids. Information dissemination in graphs that are dynamically orientable requires that number of messages sent in each direction along an edge be balanced and therefore necessitates a different approach in gossiping and multiple message broadcasting.;The obvious upper bound for gossiping and multiple message broadcasting in dynamically orientable graphs is twice the best known time for gossiping and multiple message broadcasting in classical graphs. This is obtained by inserting an additional time step t\u27 after each time step t in the classical graph algorithm in which all calls of time step t are repeated with messages moving along the same edges but in the opposite direction to reset the bias of these edges. Finding better bounds for gossiping and multiple message broadcasting in dynamically orientable graphs is the goal of this research.;For each network topology an algorithm is proposed to perform gossiping and multiple message broadcasting. For some network topologies proposed algorithms for dynamically orientable graphs achieved the same upper bound as it is known for classical graphs, for example, gossiping in dynamically orientable grid graphs. In some cases the best time is the twice the best known time for gossiping and multiple message broadcasting in classical graphs, for example, gossiping in dynamically orientable star graphs. In other cases, good time bounds are achieved that are very close to the upper bounds in classical graphs, for example, multiple message broadcasting in dynamically orientable grid graphs. Multiple message broadcasting in dynamically orientable cycle graphs is also a good example of close upper bounds. As number of messages increases bounds become very close to each other
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