The Tractability Border of Reachability in Simple Vector Addition Systems with States

Vector Addition Systems with States (VASS), equivalent to Petri nets, are a well-established model of concurrency. The central algorithmic challenge in VASS is the reachability problem: is there a run from a given starting state and counter values to a given target state and counter values? When the input is encoded in binary, reachability is computationally intractable: even in dimension one, it is NP-hard. In this paper, we comprehensively characterise the tractability border of the problem when the input is encoded in unary. For our main result, we prove that reachability is NP-hard in unary encoded 3-VASS, even when structure is heavily restricted to be a simple linear path scheme. This improves upon a recent result of Czerwi\'nski and Orlikowski (2022), in both the number of counters and expressiveness of the considered model, as well as answers open questions of Englert, Lazi\'c, and Totzke (2016) and Leroux (2021). The underlying graph structure of a simple linear path scheme (SLPS) is just a path with self-loops at each node. We also study the exceedingly weak model of computation that is SPLS with counter updates in {-1,0,+1}. Here, we show that reachability is NP-hard when the dimension is bounded by O(\alpha(k)), where \alpha is the inverse Ackermann function and k bounds the size of the SLPS. We complement our result by presenting a polynomial-time algorithm that decides reachability in 2-SLPS when the initial and target configurations are specified in binary. To achieve this, we show that reachability in such instances is well-structured: all loops, except perhaps for a constant number, are taken either polynomially many times or almost maximally. This extends the main result of Englert, Lazi\'c, and Totzke (2016) who showed the problem is in NL when the initial and target configurations are specified in unary.