Wil introduced Object-Centric Petri Nets (OCPNs), i.e., a Petri net class that more explicitly reflects the control flow of different interacting business objects. However, the theoretical properties of these nets have not been extensively studied. In this paper, we define the one-dimensional equivalent of OCPNs, i.e., Variable Arc Nets (VAR-nets), and present corresponding foundational theoretical results. VAR-nets do not support different object types; however, they do allow for the consumption and production of an arbitrary number of tokens, effectively modeling 1 : n relationships between the execution of an event and related process instances. We extend the marking equation for VAR-nets and demonstrate that this extension provides the same guarantees as the regular marking equation. Furthermore, we derive a corresponding nonlinear optimization problem and show the existence of a linear relaxation. Our experiments indicate that adopting the proposed optimization problems as a heuristic in informed state space searches speeds up the general state space search of VAR-nets significantly, compared to uninformed search algorithms. Hence, our results, based on informed search algorithms, enable feasible state space search for VAR nets.