Given $m$-dimensional vector $p$ (price vector), and $m\times n$ matrix $X$ ($m$ securities' payoffs in $n$ states) for arbitrary $m$ and $n$, is there an algorithm to decide if there exists $h\in \mathbb{R}^m$ (arbitrage portfolio) such that
$hX\geq 0\quad \mbox{and}\quad hp<0 \qquad \tag{1}$
If $X$, $p$ and $h$ can have only integer entries, can we still decide?
I know that by Farkas' Lemma that equation $(1)$ is true iff there exists $q\in \mathbb{R}^n$ such that
$ p=Xq\quad \mbox{and}\quad q\geq 0 $
But this doesn't seem to help answer the question, either.