The Surplus Parking Gathering Problem in Infinite Grids

In this paper, we introduce the \emph{Surplus Parking Gathering Problem} ($\mathcal{SPG}$), a new coordination problem for robots deployed on an infinite grid. The input consists of a set of designated parking nodes, each associated with a prescribed capacity, while the total number of robots exceeds the total parking capacity. The objective is to saturate every parking node exactly according to its capacity while gathering all remaining surplus robots at a common grid node that is not specified a priori. The robots are assumed to be autonomous, anonymous, oblivious, identical, disoriented, and homogeneous. We consider the asynchronous (\textsc{async}) model with global visibility and global strong multiplicity detection.
We first establish necessary conditions for the solvability of $\mathcal{SPG}$ by characterizing the initial configurations that admit no deterministic distributed algorithm. For all the remaining solvable configurations, we present a deterministic distributed algorithm that correctly solves the problem. The proposed algorithm proceeds in several phases and avoids collisions throughout its execution. We prove that the algorithm terminates in finite time and, upon termination, every parking node is saturated according to its prescribed capacity while all surplus robots are gathered at a uniquely determined gathering node. We further analyze the move complexity of the proposed algorithm, obtaining an $O(n(a+b)+n^2)$ upper bound together with an $\Omega(n(a+b))$ worst-case lower bound for the $\mathcal{SPG}$ problem.