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Peter Hamburger

Publications and source records attributed to Peter Hamburger.

5 recordsLinked to original sources
arXiv2020

Random bipartite posets and extremal problems

Previously, Erdős, Kierstead and Trotter investigated the dimension of random height~$2$ partially ordered sets. Their research was motivated primarily by two goals: (1)~analyzing the relative tightness of the Füredi-Kahn upper bounds on dimension in terms of maximum degree; and (2)~developing machinery for estimating the expected dimension of a random labeled poset on $n$ points. For these reasons, most of their effort was focused on the case $0<p\le 1/2$. While bounds were given for the range $1/2\le p <1$, the relative accuracy of the results in the original paper deteriorated as $p$ approaches~$1$. Motivated by two extremal problems involving conditions that force a poset to contain a large standard example, we were compelled to revisit this subject, but now with primary emphasis on the range $1/2\le p<1$. Our sharpened analysis shows that as $p$ approaches~$1$, the expected value of dimension increases and then decreases, answering in the negative a question posed in the original paper. Along the way, we apply inequalities of Talagrand and Janson, establish connections with latin rectangles and the Euler product function, and make progress on both extremal problems.

math.CO↗
arXiv2018

Double Threshold Digraphs

A semiorder is a model of preference relations where each element $x$ is associated with a utility value $α(x)$, and there is a threshold $t$ such that $y$ is preferred to $x$ iff $α(y) > α(x)+t$. These are motivated by the notion that there is some uncertainty in the utility values we assign an object or that a subject may be unable to distinguish a preference between objects whose values are close. However, they fail to model the well-known phenomenon that preferences are not always transitive. Also, if we are uncertain of the utility values, it is not logical that preference is determined absolutely by a comparison of them with an exact threshold. We propose a new model in which there are two thresholds, $t_1$ and $t_2$; if the difference $α(y) - α(x)$ less than $t_1$, then $y$ is not preferred to $x$; if the difference is greater than $t_2$ then $y$ is preferred to $x$; if it is between $t_1$ and $t_2$, then then $y$ may or may not be preferred to $x$. We call such a relation a double-threshold semiorder, and the corresponding directed graph $G = (V,E)$ a double threshold digraph. Every directed acyclic graph is a double threshold graph; bounds on $t_2/t_1$ give a nested hierarchy of subclasses of the directed acyclic graphs. In this paper we characterize the subclasses in terms of forbidden subgraphs, and give algorithms for finding an assignment of of utility values that explains the relation in terms of a given $(t_1,t_2)$ or else produces a forbidden subgraph, and finding the minimum value $λ$ of $t_2/t_1$ that is satisfiable for a given directed acyclic graph. We show that $λ$ gives a measure of the complexity of a directed acyclic graph with respect to several optimization problems that are NP-hard on arbitrary directed acyclic graphs.

cs.DS↗
arXiv2015

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset $P$, denoted $\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\le 2n+1$ is $n$, provided $n\ge2$, and this inequality is tight when $P$ contains the standard example $S_n$. However, there are posets with large dimension that do not contain the standard example $S_2$. Moreover, for each fixed $d\ge2$, if $P$ is a poset with $|P|\le 2n+1$ and $P$ does not contain the standard example $S_d$, then $\dim(P)=o(n)$. Also, for large $n$, there is a poset $P$ with $|P|=2n$ and $\dim(P)\ge (1-o(1))n$ such that the largest $d$ so that $P$ contains the standard example $S_d$ is $o(n)$. In this paper, we will show that for every integer $c\ge1$, there is an integer $f(c)=O(c^2)$ so that for large enough $n$, if $P$ is a poset with $|P|\le 2n+1$ and $\dim(P)\ge n-c$, then $P$ contains a standard example $S_d$ with $d\ge n-f(c)$. From below, we show that $f(c)=Ω(c^{4/3})$. On the other hand, we also prove an analogous result for fractional dimension, and in this setting $f(c)$ is linear in $c$. Here the result is best possible up to the value of the multiplicative constant.

math.CO↗
arXiv2014

The proof of the removable pair conjecture for fractional dimension

In 1971 Trotter conjectured that every finite poset on at least $3$ points has a pair whose removal does not decrease the dimension by more than $1$. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they made a similar conjecture for fractional dimension. This paper settles this latter conjecture.

math.CO↗