602,077 research outputs found
Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method
We use the incompressibility method based on Kolmogorov complexity to
determine the total number of bits of routing information for almost all
network topologies. In most models for routing, for almost all labeled graphs
bits are necessary and sufficient for shortest path routing. By
`almost all graphs' we mean the Kolmogorov random graphs which constitute a
fraction of of all graphs on nodes, where is an arbitrary
fixed constant. There is a model for which the average case lower bound rises
to and another model where the average case upper bound
drops to . This clearly exposes the sensitivity of such bounds
to the model under consideration. If paths have to be short, but need not be
shortest (if the stretch factor may be larger than 1), then much less space is
needed on average, even in the more demanding models. Full-information routing
requires bits on average. For worst-case static networks we
prove a lower bound for shortest path routing and all
stretch factors in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea
Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs
The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice.
We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability.
The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time
Graph Pricing Problem on Bounded Treewidth, Bounded Genus and k-partite graphs
Consider the following problem. A seller has infinite copies of products
represented by nodes in a graph. There are consumers, each has a budget and
wants to buy two products. Consumers are represented by weighted edges. Given
the prices of products, each consumer will buy both products she wants, at the
given price, if she can afford to. Our objective is to help the seller price
the products to maximize her profit.
This problem is called {\em graph vertex pricing} ({\sf GVP}) problem and has
resisted several recent attempts despite its current simple solution. This
motivates the study of this problem on special classes of graphs. In this
paper, we study this problem on a large class of graphs such as graphs with
bounded treewidth, bounded genus and -partite graphs.
We show that there exists an {\sf FPTAS} for {\sf GVP} on graphs with bounded
treewidth. This result is also extended to an {\sf FPTAS} for the more general
{\em single-minded pricing} problem. On bounded genus graphs we present a {\sf
PTAS} and show that {\sf GVP} is {\sf NP}-hard even on planar graphs.
We study the Sherali-Adams hierarchy applied to a natural Integer Program
formulation that -approximates the optimal solution of {\sf GVP}.
Sherali-Adams hierarchy has gained much interest recently as a possible
approach to develop new approximation algorithms. We show that, when the input
graph has bounded treewidth or bounded genus, applying a constant number of
rounds of Sherali-Adams hierarchy makes the integrality gap of this natural
{\sf LP} arbitrarily small, thus giving a -approximate solution
to the original {\sf GVP} instance.
On -partite graphs, we present a constant-factor approximation algorithm.
We further improve the approximation factors for paths, cycles and graphs with
degree at most three.Comment: Preprint of the paper to appear in Chicago Journal of Theoretical
Computer Scienc
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