\documentclass{article}
\usepackage{fullpage}
\pagestyle{myheadings}
\begin{document}
\section*{Column Packing Problem}
We define \(\{x_0, x_1, x_2 \ldots x_n\}_{ord}\) to be a set with an ordering
of the elements, such that \(x_0\) is before \(x_1\) is before \(x_2\)
and so on.
Consider \(c\), an arbitrary constant integer, and \(x_0, x_1, x_2
\ldots x_n\), integers such that \(\forall m: 0 < x_m \leq c\). We
wish to partition the ordered set \(\{x_0, x_1, x_2 \ldots
x_n\}_{ord}\) into ordered sets \(S_0, S_1, S_2 \ldots S_p\) such
that where \(m\) and \(p\) are such that \(n \leq pm\):
\begin{eqnarray*}
S_0 & = & \{x_0 \ldots x_{m-1}\}_{ord}\\
S_1 & = & \{x_0 \ldots x_{2m-1}\}_{ord}\\
\vdots & &\\
S_p & = & \{x_{(p-1)m} \ldots x_{n}\}_{ord}\\
\end{eqnarray*}
Define \(\lceil S_i \rceil = x_{max(S_i)}\) where \(x_{max(S_i)}\)
is the largest \(x_j\) in \(S_i\).
We wish to maximise p such that:
\[\sum_{q=0}^p \lceil S_q \rceil \leq c - p\]
A) What is the theoretical fastest time to do this.
B) Find an algorithm that takes the theoretically fastest time.
\end{document}