166 research outputs found
What Makes a Good Plan? An Efficient Planning Approach to Control Diffusion Processes in Networks
In this paper, we analyze the quality of a large class of simple dynamic
resource allocation (DRA) strategies which we name priority planning. Their aim
is to control an undesired diffusion process by distributing resources to the
contagious nodes of the network according to a predefined priority-order. In
our analysis, we reduce the DRA problem to the linear arrangement of the nodes
of the network. Under this perspective, we shed light on the role of a
fundamental characteristic of this arrangement, the maximum cutwidth, for
assessing the quality of any priority planning strategy. Our theoretical
analysis validates the role of the maximum cutwidth by deriving bounds for the
extinction time of the diffusion process. Finally, using the results of our
analysis, we propose a novel and efficient DRA strategy, called Maximum
Cutwidth Minimization, that outperforms other competing strategies in our
simulations.Comment: 18 pages, 3 figure
The Firefighter Problem: A Structural Analysis
We consider the complexity of the firefighter problem where b>=1 firefighters
are available at each time step. This problem is proved NP-complete even on
trees of degree at most three and budget one (Finbow et al.,2007) and on trees
of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this
paper, we provide further insight into the complexity landscape of the problem
by showing that the pathwidth and the maximum degree of the input graph govern
its complexity. More precisely, we first prove that the problem is NP-complete
even on trees of pathwidth at most three for any fixed budget b>=1. We then
show that the problem turns out to be fixed parameter-tractable with respect to
the combined parameter "pathwidth" and "maximum degree" of the input graph
Cutwidth: obstructions and algorithmic aspects
Cutwidth is one of the classic layout parameters for graphs. It measures how
well one can order the vertices of a graph in a linear manner, so that the
maximum number of edges between any prefix and its complement suffix is
minimized. As graphs of cutwidth at most are closed under taking
immersions, the results of Robertson and Seymour imply that there is a finite
list of minimal immersion obstructions for admitting a cut layout of width at
most . We prove that every minimal immersion obstruction for cutwidth at
most has size at most .
As an interesting algorithmic byproduct, we design a new fixed-parameter
algorithm for computing the cutwidth of a graph that runs in time , where is the optimum width and is the number of vertices.
While being slower by a -factor in the exponent than the fastest known
algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear
time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth
II: Algorithms for partial -trees of bounded degree, J. Algorithms,
56(1):25--49, 2005], our algorithm has the advantage of being simpler and
self-contained; arguably, it explains better the combinatorics of optimum-width
layouts
The Cost of Uncertainty in Curing Epidemics
Motivated by the study of controlling (curing) epidemics, we consider the
spread of an SI process on a known graph, where we have a limited budget to use
to transition infected nodes back to the susceptible state (i.e., to cure
nodes). Recent work has demonstrated that under perfect and instantaneous
information (which nodes are/are not infected), the budget required for curing
a graph precisely depends on a combinatorial property called the CutWidth. We
show that this assumption is in fact necessary: even a minor degradation of
perfect information, e.g., a diagnostic test that is 99% accurate, drastically
alters the landscape. Infections that could previously be cured in sublinear
time now may require exponential time, or orderwise larger budget to cure. The
crux of the issue comes down to a tension not present in the full information
case: if a node is suspected (but not certain) to be infected, do we risk
wasting our budget to try to cure an uninfected node, or increase our certainty
by longer observation, at the risk that the infection spreads further? Our
results present fundamental, algorithm-independent bounds that tradeoff budget
required vs. uncertainty.Comment: 35 pages, 3 figure
An efficient curing policy for epidemics on graphs
We provide a dynamic policy for the rapid containment of a contagion process
modeled as an SIS epidemic on a bounded degree undirected graph with n nodes.
We show that if the budget of curing resources available at each time is
, where is the CutWidth of the graph, and also of order
, then the expected time until the extinction of the epidemic
is of order , which is within a constant factor from optimal, as well
as sublinear in the number of nodes. Furthermore, if the CutWidth increases
only sublinearly with n, a sublinear expected time to extinction is possible
with a sublinearly increasing budget
Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number
We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard but fixed-parameter tractable (when the locality number or the alphabet size is treated as a parameter), and can be approximated with ratio O(sqrt{log{opt}} log n). As a by-product, we also relate cutwidth via the locality number to pathwidth, which is of independent interest, since it improves the best currently known approximation algorithm for cutwidth. In addition to these main results, we also consider the possibility of greedy-based approximation algorithms for the locality number
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