I am looking for an algorithm for checking the satisfiability (with natural values) of a set of inequalities made of variables and natural numbers, for example: $u < v, u \leq z, 3 \leq v$.
In this paper, http://boole.stanford.edu/pub/sefnp.pdf , there's the following claim in the second page:
"Then the conjunction is satisfiable if and only if the graph contains a cycle of positive weight, i.e., one whose labels add up to a positive integer."
The graph it mentions consists of a node for each variable in the inequalities, plus one extra node for the number $0$. Then, for each inequality, we add an edge from node $x$ to node $y$ with weight $i$ if the inequality is written as $x + i \leq y$.
Is the claim true? I can't work out any example, indeed, I think that what makes sense is exactly the opposite: the conjunction is not satisfiable if and only if the graph contains a cycle of positive weight. (it would mean that $u + k \leq u$ for some node $u$ with positive $k$).
Btw, any other algorithm regarding this problem is welcome.