9,420 research outputs found
The Power of Cut-Based Parameters for Computing Edge Disjoint Paths
This paper revisits the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3.
We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter treecut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded treecut width. Our second result shows that EDP parameterized by treecut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph
The Power of Cut-Based Parameters for Computing Edge-Disjoint Paths
This paper revisits the classical edge-disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter tree-cut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded tree-cut width. Our second result shows that EDP parameterized by tree-cut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph
A New Framework for Network Disruption
Traditional network disruption approaches focus on disconnecting or
lengthening paths in the network. We present a new framework for network
disruption that attempts to reroute flow through critical vertices via vertex
deletion, under the assumption that this will render those vertices vulnerable
to future attacks. We define the load on a critical vertex to be the number of
paths in the network that must flow through the vertex. We present
graph-theoretic and computational techniques to maximize this load, firstly by
removing either a single vertex from the network, secondly by removing a subset
of vertices.Comment: Submitted for peer review on September 13, 201
Survivability in Time-varying Networks
Time-varying graphs are a useful model for networks with dynamic connectivity
such as vehicular networks, yet, despite their great modeling power, many
important features of time-varying graphs are still poorly understood. In this
paper, we study the survivability properties of time-varying networks against
unpredictable interruptions. We first show that the traditional definition of
survivability is not effective in time-varying networks, and propose a new
survivability framework. To evaluate the survivability of time-varying networks
under the new framework, we propose two metrics that are analogous to MaxFlow
and MinCut in static networks. We show that some fundamental
survivability-related results such as Menger's Theorem only conditionally hold
in time-varying networks. Then we analyze the complexity of computing the
proposed metrics and develop several approximation algorithms. Finally, we
conduct trace-driven simulations to demonstrate the application of our
survivability framework to the robust design of a real-world bus communication
network
Universal and Robust Distributed Network Codes
Random linear network codes can be designed and implemented in a distributed
manner, with low computational complexity. However, these codes are classically
implemented over finite fields whose size depends on some global network
parameters (size of the network, the number of sinks) that may not be known
prior to code design. Also, if new nodes join the entire network code may have
to be redesigned.
In this work, we present the first universal and robust distributed linear
network coding schemes. Our schemes are universal since they are independent of
all network parameters. They are robust since if nodes join or leave, the
remaining nodes do not need to change their coding operations and the receivers
can still decode. They are distributed since nodes need only have topological
information about the part of the network upstream of them, which can be
naturally streamed as part of the communication protocol.
We present both probabilistic and deterministic schemes that are all
asymptotically rate-optimal in the coding block-length, and have guarantees of
correctness. Our probabilistic designs are computationally efficient, with
order-optimal complexity. Our deterministic designs guarantee zero error
decoding, albeit via codes with high computational complexity in general. Our
coding schemes are based on network codes over ``scalable fields". Instead of
choosing coding coefficients from one field at every node, each node uses
linear coding operations over an ``effective field-size" that depends on the
node's distance from the source node. The analysis of our schemes requires
technical tools that may be of independent interest. In particular, we
generalize the Schwartz-Zippel lemma by proving a non-uniform version, wherein
variables are chosen from sets of possibly different sizes. We also provide a
novel robust distributed algorithm to assign unique IDs to network nodes.Comment: 12 pages, 7 figures, 1 table, under submission to INFOCOM 201
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
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