19,084 research outputs found
Flexible List Colorings in Graphs with Special Degeneracy Conditions
For a given , we say that a graph is
-flexibly -choosable if the following holds: for any assignment
of color lists of size on , if a preferred color from a list is
requested at any set of vertices, then at least of these
requests are satisfied by some -coloring. We consider the question of
flexible choosability in several graph classes with certain degeneracy
conditions. We characterize the graphs of maximum degree that are
-flexibly -choosable for some , which answers a question of Dvo\v{r}\'ak, Norin, and
Postle [List coloring with requests, JGT 2019]. In particular, we show that for
any , any graph of maximum degree that is not isomorphic
to is -flexibly -choosable. Our
fraction of is within a constant factor of being the best
possible. We also show that graphs of treewidth are -flexibly
-choosable, answering a question of Choi et al.~[arXiv 2020], and we give
conditions for list assignments by which graphs of treewidth are
-flexibly -choosable. We show furthermore that graphs of
treedepth are -flexibly -choosable. Finally, we introduce a
notion of flexible degeneracy, which strengthens flexible choosability, and we
show that apart from a well-understood class of exceptions, 3-connected
non-regular graphs of maximum degree are flexibly -degenerate.Comment: 21 pages, 5 figure
Adding Isolated Vertices Makes some Online Algorithms Optimal
An unexpected difference between online and offline algorithms is observed.
The natural greedy algorithms are shown to be worst case online optimal for
Online Independent Set and Online Vertex Cover on graphs with 'enough' isolated
vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is
shown to be worst case online optimal on graphs with at least one isolated
vertex. These algorithms are not online optimal in general. The online
optimality results for these greedy algorithms imply optimality according to
various worst case performance measures, such as the competitive ratio. It is
also shown that, despite this worst case optimality, there are Freckle graphs
where the greedy independent set algorithm is objectively less good than
another algorithm. It is shown that it is NP-hard to determine any of the
following for a given graph: the online independence number, the online vertex
cover number, and the online domination number.Comment: A footnote in the .tex file didn't show up in the last version. This
was fixe
Coded Caching for Delay-Sensitive Content
Coded caching is a recently proposed technique that achieves significant
performance gains for cache networks compared to uncoded caching schemes.
However, this substantial coding gain is attained at the cost of large delivery
delay, which is not tolerable in delay-sensitive applications such as video
streaming. In this paper, we identify and investigate the tradeoff between the
performance gain of coded caching and the delivery delay. We propose a
computationally efficient caching algorithm that provides the gains of coding
and respects delay constraints. The proposed algorithm achieves the optimum
performance for large delay, but still offers major gains for small delay.
These gains are demonstrated in a practical setting with a video-streaming
prototype.Comment: 9 page
Online Multi-Coloring with Advice
We consider the problem of online graph multi-coloring with advice.
Multi-coloring is often used to model frequency allocation in cellular
networks. We give several nearly tight upper and lower bounds for the most
standard topologies of cellular networks, paths and hexagonal graphs. For the
path, negative results trivially carry over to bipartite graphs, and our
positive results are also valid for bipartite graphs. The advice given
represents information that is likely to be available, studying for instance
the data from earlier similar periods of time.Comment: IMADA-preprint-c
Grafalgo - A Library of Graph Algorithms and Supporting Data Structures (revised)
This report provides an (updated) overview of {\sl Grafalgo}, an open-source
library of graph algorithms and the data structures used to implement them. The
programs in this library were originally written to support a graduate class in
advanced data structures and algorithms at Washington University. Because the
code's primary purpose was pedagogical, it was written to be as straightforward
as possible, while still being highly efficient. Grafalgo is implemented in C++
and incorporates some features of C++11.
The library is available on an open-source basis and may be downloaded from
https://code.google.com/p/grafalgo/. Source code documentation is at
www.arl.wustl.edu/\textasciitilde jst/doc/grafalgo. While not designed as
production code, the library is suitable for use in larger systems, so long as
its limitations are understood. The readability of the code also makes it
relatively straightforward to extend it for other purposes
An Efficient Coded Multicasting Scheme Preserving the Multiplicative Caching Gain
Coded multicasting has been shown to be a promis- ing approach to
significantly improve the caching performance of content delivery networks with
multiple caches downstream of a common multicast link. However, achievable
schemes proposed to date have been shown to achieve the proved order-optimal
performance only in the asymptotic regime in which the number of packets per
requested item goes to infinity. In this paper, we first extend the asymptotic
analysis of the achievable scheme in [1], [2] to the case of heterogeneous
cache sizes and demand distributions, providing the best known upper bound on
the fundamental limiting performance when the number of packets goes to
infinity. We then show that the scheme achieving this upper bound quickly loses
its multiplicative caching gain for finite content packetization. To overcome
this limitation, we design a novel polynomial-time algorithm based on random
greedy graph- coloring that, while keeping the same finite content
packetization, recovers a significant part of the multiplicative caching gain.
Our results show that the order-optimal coded multicasting schemes proposed to
date, while useful in quantifying the fundamental limiting performance, must be
properly designed for practical regimes of finite packetization.Comment: 6 pages, 7 figures, Published in Infocom CNTCV 201
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