103 research outputs found
The complexity of counting poset and permutation patterns
We introduce a notion of pattern occurrence that generalizes both classical
permutation patterns as well as poset containment. Many questions about pattern
statistics and avoidance generalize naturally to this setting, and we focus on
functional complexity problems -- particularly those that arise by constraining
the order dimensions of the pattern and text posets. We show that counting the
number of induced, injective occurrences among dimension 2 posets is #P-hard;
enumerating the linear extensions that occur in realizers of dimension 2 posets
can be done in polynomial time, while for unconstrained dimension it is
GI-complete; counting not necessarily induced, injective occurrences among
dimension 2 posets is #P-hard; counting injective or not necessarily injective
occurrences of an arbitrary pattern in a dimension 1 text is #P-hard, although
it is in FP if the pattern poset is constrained to have bounded intrinsic
width; and counting injective occurrences of a dimension 1 pattern in an
arbitrary text is #P-hard, while it is in FP for bounded dimension texts. This
framework easily leads to a number of open questions, chief among which are (1)
is it #P-hard to count the number of occurrences of a dimension 2 pattern in a
dimension 1 text, and (2) is it #P-hard to count the number of texts which
avoid a given pattern?Comment: 15 page
Local Boxicity, Local Dimension, and Maximum Degree
In this paper, we focus on two recently introduced parameters in the
literature, namely `local boxicity' (a parameter on graphs) and `local
dimension' (a parameter on partially ordered sets). We give an `almost linear'
upper bound for both the parameters in terms of the maximum degree of a graph
(for local dimension we consider the comparability graph of a poset). Further,
we give an time deterministic algorithm to compute a local box
representation of dimension at most for a claw-free graph, where
and denote the number of vertices and the maximum degree,
respectively, of the graph under consideration. We also prove two other upper
bounds for the local boxicity of a graph, one in terms of the number of
vertices and the other in terms of the number of edges. Finally, we show that
the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page
SOME ASPECTS OF TOPOLOGICAL SORTING
In this paper, we provide an outline of most of the known techniques and principal results pertaining to computing and counting topological sorts, realizers and dimension of a finite partially ordered set, and identify some new directions
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
Let be an -node planar graph. In a visibility representation of ,
each node of is represented by a horizontal line segment such that the line
segments representing any two adjacent nodes of are vertically visible to
each other. In the present paper we give the best known compact visibility
representation of . Given a canonical ordering of the triangulated , our
algorithm draws the graph incrementally in a greedy manner. We show that one of
three canonical orderings obtained from Schnyder's realizer for the
triangulated yields a visibility representation of no wider than
. Our easy-to-implement O(n)-time algorithm bypasses the
complicated subroutines for four-connected components and four-block trees
required by the best previously known algorithm of Kant. Our result provides a
negative answer to Kant's open question about whether is a
worst-case lower bound on the required width. Also, if has no degree-three
(respectively, degree-five) internal node, then our visibility representation
for is no wider than (respectively, ).
Moreover, if is four-connected, then our visibility representation for
is no wider than , matching the best known result of Kant and He. As a
by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem
on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to
appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of
Computer Science (STACS), Berlin, Germany, 200
Dimension of posets with planar cover graphs excluding two long incomparable chains
It has been known for more than 40 years that there are posets with planar
cover graphs and arbitrarily large dimension. Recently, Streib and Trotter
proved that such posets must have large height. In fact, all known
constructions of such posets have two large disjoint chains with all points in
one chain incomparable with all points in the other. Gutowski and Krawczyk
conjectured that this feature is necessary. More formally, they conjectured
that for every , there is a constant such that if is a poset
with a planar cover graph and excludes , then
. We settle their conjecture in the affirmative. We also discuss
possibilities of generalizing the result by relaxing the condition that the
cover graph is planar.Comment: New section on connections with graph minors, small correction
Learning the String Partial Order
We show that most structured prediction problems can be solved in linear time
and space by considering them as partial orderings of the tokens in the input
string. Our method computes real numbers for each token in an input string and
sorts the tokens accordingly, resulting in as few as 2 total orders of the
tokens in the string. Each total order possesses a set of edges oriented from
smaller to greater tokens. The intersection of total orders results in a
partial order over the set of input tokens, which is then decoded into a
directed graph representing the desired structure. Experiments show that our
method achieves 95.4 LAS and 96.9 UAS by using an intersection of 2 total
orders, 95.7 LAS and 97.1 UAS with 4 on the English Penn Treebank dependency
parsing benchmark. Our method is also the first linear-complexity coreference
resolution model and achieves 79.2 F1 on the English OntoNotes benchmark, which
is comparable with state of the art.Comment: 12 page
- …