A 1-dimensional Vector Addition System with States (1-VASS) can be seen as an directed and integer weighted graph that is equipped with a non-negative integer counter.

A configuration of a 1-VASS is a pair $(q, x)$ that is the current node $q$ and the current non-negative counter value $x$.

A run in a 1-VASS is a sequence of configurations $(q_0, x_0), (q_1, x_1), \ldots, (q_k, x_k)$ where for each $i \in \{ 1, \ldots, k\}$, there is an edge from $x_{i-1}$ to $x_i$ with weight $x_i - x_{i-1}$.

Let $n$ the size of a 1-VASS, encoded in unary, that is the number of states plus the absolute value of all weights.

The coverability problem asks: in a given 1-VASS, is there is a run from a given starting state that ends in a given target state (with any counter value).

There is a straightforward $O(n^{2})$-time algorithm for coverability in 1-VASS.

Does is there an $o(n^{2})$-time algorithm for coverability in 1-VASS?