Coverability in VASS Revisited: Improving Rackoff’s Bound to Obtain Conditional Optimality

Seminal results establish that the coverability problem for Vector Addition Systems with States (VASS) is in EXPSPACE (Rackoff, '78) and is EXPSPACE-hard already under unary encodings (Lipton, '76). More precisely, Rosier and Yen later utilise Rackoff’s bounding technique to show that if coverability holds then there is a run of length at most $n^{2^𝒪(d log d)}$, where d is the dimension and n is the size of the given unary VASS. Earlier, Lipton showed that there exist instances of coverability in d-dimensional unary VASS that are only witnessed by runs of length at least $n^{2^Ω(d)}$. Our first result closes this gap. We improve the upper bound by removing the twice-exponentiated log(d) factor, thus matching Lipton’s lower bound. This closes the corresponding gap for the exact space required to decide coverability. This also yields a deterministic $n^{2^𝒪(d)}$-time algorithm for coverability. Our second result is a matching lower bound, that there does not exist a deterministic $n^{2^o(d)}$-time algorithm, conditioned upon the Exponential Time Hypothesis.
When analysing coverability, a standard proof technique is to consider VASS with bounded counters. ... mehrBounded VASS make for an interesting and popular model due to strong connections with timed automata. Withal, we study a natural setting where the counter bound is linear in the size of the VASS. Here the trivial exhaustive search algorithm runs in $𝒪(n^{d+1})$-time. We give evidence to this being near-optimal. We prove that in dimension one this trivial algorithm is conditionally optimal, by showing that $n^{2-o(1)}$-time is required under the k-cycle hypothesis. In general fixed dimension d, we show that $n^{d-2-o(1)}$-time is required under the 3-uniform hyperclique hypothesis.