Coverability in VASS Revisited: Improving Rackoff's Bound to Obtain Conditional Optimality.
Marvin Künnemann, Filip Mazowiecki, Lia Schütze, Henry Sinclair-Banks, and Karol Węgrzycki.
In ICALP'23.
DOI: https://doi.org/10.4230/LIPIcs.ICALP.2023.131.
Full version: https://arxiv.org/abs/2305.01581.
Abstract: 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.