1,045 research outputs found
Replica Placement on Bounded Treewidth Graphs
We consider the replica placement problem: given a graph with clients and
nodes, place replicas on a minimum set of nodes to serve all the clients; each
client is associated with a request and maximum distance that it can travel to
get served and there is a maximum limit (capacity) on the amount of request a
replica can serve. The problem falls under the general framework of capacitated
set covering. It admits an O(\log n)-approximation and it is NP-hard to
approximate within a factor of . We study the problem in terms of
the treewidth of the graph and present an O(t)-approximation algorithm.Comment: An abridged version of this paper is to appear in the proceedings of
WADS'1
A simple dual ascent algorithm for the multilevel facility location problem
We present a simple dual ascent method for the multilevel facility location problem which finds a solution within times the optimum for the uncapacitated case and within times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique. \u
Distributed Connectivity Decomposition
We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
(I) A decomposition of each undirected graph with vertex-connectivity
into (fractionally) vertex-disjoint weighted dominating trees with total weight
, in rounds.
(II) A decomposition of each undirected graph with edge-connectivity
into (fractionally) edge-disjoint weighted spanning trees with total
weight , in
rounds.
We also show round complexity lower bounds of
and
for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from
to near-optimal .
As corollaries, we also get distributed oblivious routing broadcast with
-competitive edge-congestion and -competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal -approximation of vertex connectivity: centralized
and distributed . The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation
Capacitated Trees, Capacitated Routing, and Associated Polyhedra
We study the polyhedral structure of two related core combinatorial problems: the subtree cardinalityconstrained minimal spanning tree problem and the identical customer vehicle routing problem. For each of these problems, and for a forest relaxation of the minimal spanning tree problem, we introduce a number of new valid inequalities and specify conditions for ensuring when these inequalities are facets for the associated integer polyhedra. The inequalities are defined by one of several underlying support graphs: (i) a multistar, a "star" with a clique replacing the central vertex; (ii) a clique cluster, a collection of cliques intersecting at a single vertex, or more generally at a central" clique; and (iii) a ladybug, consisting of a multistar as a head and a clique as a body. We also consider packing (generalized subtour elimination) constraints, as well as several variants of our basic inequalities, such as partial multistars, whose satellite vertices need not be connected to all of the central vertices. Our development highlights the relationship between the capacitated tree and capacitated forest polytopes and a so-called path-partitioning polytope,and shows how to use monotone polytopes and a set of simple exchange arguments to prove that valid inequalities are facets
Cut Tree Construction from Massive Graphs
The construction of cut trees (also known as Gomory-Hu trees) for a given
graph enables the minimum-cut size of the original graph to be obtained for any
pair of vertices. Cut trees are a powerful back-end for graph management and
mining, as they support various procedures related to the minimum cut, maximum
flow, and connectivity. However, the crucial drawback with cut trees is the
computational cost of their construction. In theory, a cut tree is built by
applying a maximum flow algorithm for times, where is the number of
vertices. Therefore, naive implementations of this approach result in cubic
time complexity, which is obviously too slow for today's large-scale graphs. To
address this issue, in the present study, we propose a new cut-tree
construction algorithm tailored to real-world networks. Using a series of
experiments, we demonstrate that the proposed algorithm is several orders of
magnitude faster than previous algorithms and it can construct cut trees for
billion-scale graphs.Comment: Short version will appear at ICDM'1
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