Presburger arithmetic with stars, rational subsets of graph groups, and nested zero tests

We study the computational complexity of existential Presburger arithmetic with (possibly nested occurrences of) a Kleene-star operator. In addition to being a natural extension of Presburger arithmetic, our investigation is motivated by two other decision problems. The first problem is the rational subset membership problem in graph groups. A graph group is an infinite group specified by a finite undirected graph. While a characterisation of graph groups with a decidable rational subset membership problem was given by Lohrey and Steinberg [J. Algebra, 320(2) (2008)], it has been an open problem (i) whether the decidable fragment has elementary complexity and (ii) what is the complexity for each fixed graph group. The second problem is the reachability problem for integer vector addition systems with states and nested zero tests. We prove that the satisfiability problem for existential Pres-burger arithmetic with stars is NEXP-complete and that all three problems are polynomially inter-reducible. Moreover, we consider for each problem a variant with a fixed parameter: We fix the star-height in the logic, a graph parameter for the membership problem, and the number of distinct zero-tests in the integer vector addition systems. We establish NP-completeness of all problems with fixed parameters. In particular, this enables us to obtain a complete description of the complexity landscape of the rational subset membership problem for fixed graph groups: If the graph is a clique, the problem is N L-complete. If the graph is a disjoint union of cliques, it is P-complete. If it is a transitive forest (and not a union of cliques), the problem is NP-complete. Otherwise, the problem is undecidable.