  We consider parity games on infinite graphs where configurations are
represented by control-states and integer vectors. This framework subsumes two
classic game problems: parity games on vector addition systems with states
(vass) and multidimensional energy parity games. We show that the
multidimensional energy parity game problem is inter-reducible with a subclass
of single-sided parity games on vass where just one player can modify the
integer counters and the opponent can only change control-states. Our main
result is that the minimal elements of the upward-closed winning set of these
single-sided parity games on vass are computable. This implies that the Pareto
frontier of the minimal initial credit needed to win multidimensional energy
parity games is also computable, solving an open question from the literature.
Moreover, our main result implies the decidability of weak simulation
preorder/equivalence between finite-state systems and vass, and the
decidability of model checking vass with a large fragment of the modal
mu-calculus.
