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 $O(n\Delta^2)$ time deterministic algorithm to compute a local box representation of dimension at most $3\Delta$ for a claw-free graph, where $n$ and $\Delta$ 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 $G$ be an $n$-node planar graph. In a visibility representation of $G$, each node of $G$ is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of $G$ are vertically visible to each other. In the present paper we give the best known compact visibility representation of $G$. Given a canonical ordering of the triangulated $G$, 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 $G$ yields a visibility representation of $G$ no wider than $\frac{22n-40}{15}$. 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 $\frac{3n-6}{2}$ is a worst-case lower bound on the required width. Also, if $G$ has no degree-three (respectively, degree-five) internal node, then our visibility representation for $G$ is no wider than $\frac{4n-9}{3}$ (respectively, $\frac{4n-7}{3}$). Moreover, if $G$ is four-connected, then our visibility representation for $G$ is no wider than $n-1$, 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 $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. 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