51 research outputs found
On-line partitioning of width w posets into w^O(log log w) chains
An on-line chain partitioning algorithm receives the elements of a poset one
at a time, and when an element is received, irrevocably assigns it to one of
the chains. In this paper, we present an on-line algorithm that partitions
posets of width into chains. This improves over
previously best known algorithms using chains by Bosek and
Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm
runs in time, where is the width and is the size of
a presented poset.Comment: 16 pages, 10 figure
Majority choosability of digraphs
A \emph{majority coloring} of a digraph is a coloring of its vertices such
that for each vertex , at most half of the out-neighbors of has the same
color as . A digraph is \emph{majority -choosable} if for any
assignment of lists of colors of size to the vertices there is a majority
coloring of from these lists. We prove that every digraph is majority
-choosable. This gives a positive answer to a question posed recently by
Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain
this result as a consequence of a more general theorem, in which majority
condition is profitably extended. For instance, the theorem implies also that
every digraph has a coloring from arbitrary lists of size three, in which at
most of the out-neighbors of any vertex share its color. This solves
another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture
stating that every digraph is majority -colorable
A Tight Bound for Shortest Augmenting Paths on Trees
The shortest augmenting path technique is one of the fundamental ideas used
in maximum matching and maximum flow algorithms. Since being introduced by
Edmonds and Karp in 1972, it has been widely applied in many different
settings. Surprisingly, despite this extensive usage, it is still not well
understood even in the simplest case: online bipartite matching problem on
trees. In this problem a bipartite tree is being revealed
online, i.e., in each round one vertex from with its incident edges
arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis,
R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with
augmentations. In INFOCOM 2009] that the total length of all shortest
augmenting paths found is . In this paper, we prove a tight upper bound for the total length of shortest augmenting paths for
trees improving over bound [B. Bosek, D. Leniowski, P.
Sankowski, and A. Zych. Shortest augmenting paths for online matchings on
trees. In WAOA 2015].Comment: 22 pages, 10 figure
Local dimension is unbounded for planar posets
In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height
Centroidal localization game
One important problem in a network is to locate an (invisible) moving entity
by using distance-detectors placed at strategical locations. For instance, the
metric dimension of a graph is the minimum number of detectors placed
in some vertices such that the vector
of the distances between the detectors and the entity's location
allows to uniquely determine . In a more realistic setting, instead
of getting the exact distance information, given devices placed in
, we get only relative distances between the entity's
location and the devices (for every , it is provided
whether , , or to ). The centroidal dimension of a
graph is the minimum number of devices required to locate the entity in
this setting.
We consider the natural generalization of the latter problem, where vertices
may be probed sequentially until the moving entity is located. At every turn, a
set of vertices is probed and then the relative distances
between the vertices and the current location of the entity are
given. If not located, the moving entity may move along one edge. Let be the minimum such that the entity is eventually located, whatever it
does, in the graph .
We prove that for every tree and give an upper bound
on in cartesian product of graphs and . Our main
result is that for any outerplanar graph . We then prove
that is bounded by the pathwidth of plus 1 and that the
optimization problem of determining is NP-hard in general graphs.
Finally, we show that approximating (up to any constant distance) the entity's
location in the Euclidean plane requires at most two vertices per turn
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