Coverability in VASS Revisited: Improving Rackoff's Bound to Obtain Conditional Optimality
share
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. 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(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.
READ FULL TEXT