442 research outputs found
Beyond Geometry : Towards Fully Realistic Wireless Models
Signal-strength models of wireless communications capture the gradual fading
of signals and the additivity of interference. As such, they are closer to
reality than other models. However, nearly all theoretic work in the SINR model
depends on the assumption of smooth geometric decay, one that is true in free
space but is far off in actual environments. The challenge is to model
realistic environments, including walls, obstacles, reflections and anisotropic
antennas, without making the models algorithmically impractical or analytically
intractable.
We present a simple solution that allows the modeling of arbitrary static
situations by moving from geometry to arbitrary decay spaces. The complexity of
a setting is captured by a metricity parameter Z that indicates how far the
decay space is from satisfying the triangular inequality. All results that hold
in the SINR model in general metrics carry over to decay spaces, with the
resulting time complexity and approximation depending on Z in the same way that
the original results depends on the path loss term alpha. For distributed
algorithms, that to date have appeared to necessarily depend on the planarity,
we indicate how they can be adapted to arbitrary decay spaces.
Finally, we explore the dependence on Z in the approximability of core
problems. In particular, we observe that the capacity maximization problem has
exponential upper and lower bounds in terms of Z in general decay spaces. In
Euclidean metrics and related growth-bounded decay spaces, the performance
depends on the exact metricity definition, with a polynomial upper bound in
terms of Z, but an exponential lower bound in terms of a variant parameter phi.
On the plane, the upper bound result actually yields the first approximation of
a capacity-type SINR problem that is subexponential in alpha
Semi-Transitive Orientations and Word-Representable Graphs
A graph is a \emph{word-representable graph} if there exists a word
over the alphabet such that letters and alternate in if and
only if for each .
In this paper we give an effective characterization of word-representable
graphs in terms of orientations. Namely, we show that a graph is
word-representable if and only if it admits a \emph{semi-transitive
orientation} defined in the paper. This allows us to prove a number of results
about word-representable graphs, in particular showing that the recognition
problem is in NP, and that word-representable graphs include all 3-colorable
graphs.
We also explore bounds on the size of the word representing the graph. The
representation number of is the minimum such that is a
representable by a word, where each letter occurs times; such a exists
for any word-representable graph. We show that the representation number of a
word-representable graph on vertices is at most , while there exist
graphs for which it is .Comment: arXiv admin note: text overlap with arXiv:0810.031
Strongly simplicial vertices of powers of trees
AbstractFor a tree T and an integer k⩾1, it is well known that the kth power Tk of T is strongly chordal and hence has a strong elimination ordering of its vertices. In this note we obtain a complete characterization of strongly simplicial vertices of Tk, thereby characterizing all strong elimination orderings of the vertices of Tk
Independent sets in bounded-degree hypergraphs
AbstractIn this paper we analyze several approaches to the Maximum Independent Set (MIS) problem in hypergraphs with degree bounded by a parameter Δ. Since independent sets in hypergraphs can be strong and weak, we denote by MIS (MSIS) the problem of finding a maximum weak (strong) independent set in hypergraphs, respectively. We propose a general technique that reduces the worst case analysis of certain algorithms on hypergraphs to their analysis on ordinary graphs. This technique allows us to show that the greedy algorithm for MIS that corresponds to the classical greedy set cover algorithm has a performance ratio of (Δ+1)/2. It also allows us to apply results on local search algorithms on graphs to obtain a (Δ+1)/2 approximation for the weighted MIS and (Δ+3)/5−ϵ approximation for the unweighted case. We improve the bound in the weighted case to ⌈(Δ+1)/3⌉ using a simple partitioning algorithm. We also consider another natural greedy algorithm for MIS that adds vertices of minimum degree and achieves only a ratio of Δ−1, significantly worse than on ordinary graphs. For MSIS, we give two variations of the basic greedy algorithm and describe a family of hypergraphs where both algorithms approach the bound of Δ
Deterministic Digital Clustering of Wireless Ad Hoc Networks
We consider deterministic distributed communication in wireless ad hoc
networks of identical weak devices under the SINR model without predefined
infrastructure. Most algorithmic results in this model rely on various
additional features or capabilities, e.g., randomization, access to geographic
coordinates, power control, carrier sensing with various precision of
measurements, and/or interference cancellation. We study a pure scenario, when
no such properties are available. As a general tool, we develop a deterministic
distributed clustering algorithm. Our solution relies on a new type of
combinatorial structures (selectors), which might be of independent interest.
Using the clustering, we develop a deterministic distributed local broadcast
algorithm accomplishing this task in rounds, where
is the density of the network. To the best of our knowledge, this is
the first solution in pure scenario which is only polylog away from the
universal lower bound , valid also for scenarios with
randomization and other features. Therefore, none of these features
substantially helps in performing the local broadcast task. Using clustering,
we also build a deterministic global broadcast algorithm that terminates within
rounds, where is the diameter of the
network. This result is complemented by a lower bound , where is the path-loss parameter of the
environment. This lower bound shows that randomization or knowledge of own
location substantially help (by a factor polynomial in ) in the global
broadcast. Therefore, unlike in the case of local broadcast, some additional
model features may help in global broadcast
Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems
When ties and incomplete preference lists are permitted in the Stable Marriage and Hospitals/Residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe polynomial-time 5/3-approximation algorithms for variants of these problems in which ties are on one side only and at the end of the preference lists. The particular variant is motivated by important applications in large scale centralised matching schemes
Distributed Approximation of Maximum Independent Set and Maximum Matching
We present a simple distributed -approximation algorithm for maximum
weight independent set (MaxIS) in the model which completes
in rounds, where is the maximum
degree, is the number of rounds needed to compute a maximal
independent set (MIS) on , and is the maximum weight of a node. %Whether
our algorithm is randomized or deterministic depends on the \texttt{MIS}
algorithm used as a black-box.
Plugging in the best known algorithm for MIS gives a randomized solution in
rounds, where is the number of nodes.
We also present a deterministic -round algorithm based
on coloring.
We then show how to use our MaxIS approximation algorithms to compute a
-approximation for maximum weight matching without incurring any additional
round penalty in the model. We use a known reduction for
simulating algorithms on the line graph while incurring congestion, but we show
our algorithm is part of a broad family of \emph{local aggregation algorithms}
for which we describe a mechanism that allows the simulation to run in the
model without an additional overhead.
Next, we show that for maximum weight matching, relaxing the approximation
factor to () allows us to devise a distributed algorithm
requiring rounds for any constant
. For the unweighted case, we can even obtain a
-approximation in this number of rounds. These algorithms are
the first to achieve the provably optimal round complexity with respect to
dependency on
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