684 research outputs found
Packing Steiner Trees
Let be a distinguished subset of vertices in a graph . A
-\emph{Steiner tree} is a subgraph of that is a tree and that spans .
Kriesell conjectured that contains pairwise edge-disjoint -Steiner
trees provided that every edge-cut of that separates has size .
When a -Steiner tree is a spanning tree and the conjecture is a
consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved
that Kriesell's conjecture holds when is replaced by , and recently
West and Wu have lowered this value to . Our main result makes a further
improvement to .Comment: 38 pages, 4 figure
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 %
Hardness results and approximation algorithms for some problems on graphs
This thesis has two parts. In the first part, we study some graph covering problems with a non-local covering rule that allows a "remote" node to be covered by repeatedly applying the covering rule. In the second part, we provide some results on the packing of Steiner trees.
In the Propagation problem we are given a graph and the goal is to find a minimum-sized set of nodes that covers all of the nodes, where a node is covered if (1) is in , or (2) has a neighbor such that and all of its neighbors except are covered. Rule (2) is called the propagation rule, and it is applied iteratively. Throughout, we use to denote the number of nodes in the input graph. We prove that the path-width parameter is a lower bound for the optimal value. We show that the Propagation problem is NP-hard in planar weighted graphs. We prove that it is NP-hard to approximate the optimal value to within a factor of in weighted (general) graphs.
The second problem that we study is the Power Dominating Set problem. This problem has two covering rules. The first rule is the same as the domination rule as in the Dominating Set problem, and the second rule is the same propagation rule as in the Propagation problem.
We show that it is hard to approximate the optimal value to within a factor of in general graphs. We design and analyze an approximation algorithm with a performance guarantee of on planar graphs.
We formulate a common generalization of the above two problems called the General Propagation problem. We reformulate this general problem as an orientation problem, and based on this reformulation we design a dynamic programming algorithm. The algorithm runs in linear time when the graph has tree-width . Motivated by applications, we introduce a restricted version of the problem that we call the -round General Propagation problem. We give a PTAS for the -round General Propagation problem on planar graphs, for small values of . Our dynamic programming algorithms and the PTAS can be extended to other problems in networks with similar propagation rules. As an example we discuss the extension of our results to the Target Set Selection problem in the threshold model of the diffusion processes.
In the second part of the thesis, we focus on the Steiner Tree Packing problem. In this problem, we are given a graph and a subset of terminal nodes . The goal in this problem is to find a maximum cardinality set of disjoint trees that each spans , that is, each of the trees should contain all terminal nodes. In the edge-disjoint version of this problem, the trees have to be edge disjoint. In the element-disjoint version, the trees have to be node disjoint on non-terminal nodes and edge-disjoint on edges adjacent to terminals. We show that both problems are NP-hard when there are only terminals. Our main focus is on planar instances of these problems. We show that the edge-disjoint version of the problem is NP-hard even in planar graphs with terminals on the same face of the embedding. Next, we design an algorithm that achieves an approximation guarantee of , given a planar graph that is element-connected on the terminals; in fact, given such a graph the algorithm returns element-disjoint Steiner trees. Using this algorithm we get an approximation algorithm with guarantee of (almost) for the edge-disjoint version of the problem in planar graphs. We also show that the natural LP relaxation of the edge-disjoint Steiner Tree Packing problem has an integrality ratio
of in planar graphs
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