13,005 research outputs found
The 2-hop spanning tree problem
Given a graph G with a specified root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2-hop spanning tree. For given edge weights the 2-hop spanning tree problem is to find a minimum weight 2-hop spanning tree. The problem is NP-hard and has some interesting applications. We study a polytope associated with a directed model of the problem give a completeness result for wheels and a vertex description of a linear relaxation. Some classes of valid inequalities for the convex hull of incidence vectors of 2-hop spanning trees are derived by projection techniques
Improving Robustness of Next-Hop Routing
A weakness of next-hop routing is that following a link or router failure
there may be no routes between some source-destination pairs, or packets may
get stuck in a routing loop as the protocol operates to establish new routes.
In this article, we address these weaknesses by describing mechanisms to choose
alternate next hops.
Our first contribution is to model the scenario as the following {\sc tree
augmentation} problem. Consider a mixed graph where some edges are directed and
some undirected. The directed edges form a spanning tree pointing towards the
common destination node. Each directed edge represents the unique next hop in
the routing protocol. Our goal is to direct the undirected edges so that the
resulting graph remains acyclic and the number of nodes with outdegree two or
more is maximized. These nodes represent those with alternative next hops in
their routing paths.
We show that {\sc tree augmentation} is NP-hard in general and present a
simple -approximation algorithm. We also study 3 special cases. We
give exact polynomial-time algorithms for when the input spanning tree consists
of exactly 2 directed paths or when the input graph has bounded treewidth. For
planar graphs, we present a polynomial-time approximation scheme when the input
tree is a breadth-first search tree. To the best of our knowledge, {\sc tree
augmentation} has not been previously studied
Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty
We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral ? ? 2, we present algorithms that are ?-robust and (1+1/?)-consistent, meaning that they use at most ?OPT queries if the predictions are arbitrarily wrong and at most (1+1/?)OPT queries if the predictions are correct, where OPT is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of 2, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets - the key to lower-bounding the optimum - by proposing novel global witness set structures and completely new ways of adaptively using those
New formulations for the hop-constrained minimum spanning tree problem via Sherali and Driscoll's tightened Miller-Tucker-Zemlin constraints
Given an undirected network with positive edge costs and a natural number p, the hop-constrained minimum spanning tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, the new models based on the Miller-Tucker-Zemlin (MTZ) subtour elimination constraints are developed and computational results together with comparisons against MTZ-based, flow-based, and hop-indexed formulations are reported. The first model is obtained by adapting the MTZ-based Asymmetric Traveling Salesman Problem formulation of Sherali and Driscoll [18] and the other two models are obtained by combining topology-enforcing and MTZ-related constraints offered by Akgün and Tansel (submitted for publication) [20] for HMST with the first model appropriately. Computational studies show that the best LP bounds of the MTZ-based models in the literature are improved by the proposed models. The best solution times of the MTZ-based models are not improved for optimally solved instances. However, the results for the harder, large-size instances imply that the proposed models are likely to produce better solution times. The proposed models do not dominate the flow-based and hop-indexed formulations with respect to LP bounds. However, good feasible solutions can be obtained in a reasonable amount of time for problems for which even the LP relaxations of the flow-based and hop-indexed formulations can be solved in about 2 days. © 2010 Elsevier Ltd. All rights reserved
Power Aware Routing for Sensor Databases
Wireless sensor networks offer the potential to span and monitor large
geographical areas inexpensively. Sensor network databases like TinyDB are the
dominant architectures to extract and manage data in such networks. Since
sensors have significant power constraints (battery life), and high
communication costs, design of energy efficient communication algorithms is of
great importance. The data flow in a sensor database is very different from
data flow in an ordinary network and poses novel challenges in designing
efficient routing algorithms. In this work we explore the problem of energy
efficient routing for various different types of database queries and show that
in general, this problem is NP-complete. We give a constant factor
approximation algorithm for one class of query, and for other queries give
heuristic algorithms. We evaluate the efficiency of the proposed algorithms by
simulation and demonstrate their near optimal performance for various network
sizes
Single failure resiliency in greedy routing
Using greedy routing, network nodes forward packets towards neighbors which are closer to their destination. This approach makes greedy routers significantly more memory-efficient than traditional IP-routers using longest-prefix matching. Greedy embeddings map network nodes to coordinates, such that greedy routing always leads to the destination. Prior works showed that using a spanning tree of the network topology, greedy embeddings can be found in different metric spaces for any graph. However, a single link/node failure might affect the greedy embedding and causes the packets to reach a dead end. In order to cope with network failures, existing greedy methods require large resources and cause significant loss in the quality of the routing (stretch loss). We propose efficient recovery techniques which require very limited resources with minor effect on the stretch. As the proposed techniques are protection, the switch-over takes place very fast. Low overhead, simplicity and scalability of the methods make them suitable for large-scale networks. The proposed schemes are validated on large topologies with properties similar to the Internet. The performances of the schemes are compared with an existing alternative referred as gravity pressure routing
- …