Coverability in VASS Revisited: Improving Rackoff’s Bounds 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 \(n^{2O(d log (d))} \) , 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 \(n^{2Ω (d)} \) . 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 \(n^{2O(d)} \) -time algorithm for coverability. Our second result is a matching lower bound, that there does not exist a deterministic \(n^{2ᵒ⁽ᵈ⁾} \) -time algorithm, conditioned upon the exponential time hypothesis. When analysing coverability, a standard proof technique is to consider VASS with bounded counters. Bounded 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 \(O(nᵈ⁺¹) \) 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, for any fixed dimension d ≥ 4, we show that n d − 2 − o (1) time is required under the 3-uniform hyperclique hypothesis.