Expected Cost of Greedy Online Facility Assignment on Regular Polygons (v3)

We study a greedy online facility assignment process on a regular $n$-gon, where unit-capacity facilities occupy the vertices and customers arrive sequentially at uniformly random locations on polygon edges. Each arrival is irrevocably assigned to the nearest currently free facility under the shortest edge-walk metric, with uniform tie-breaking among equidistant choices. Our main theoretical result is an exact value-function characterization: for every occupancy state $S\subseteq V$, the expected remaining cost $V(S)$ satisfies a finite-horizon integral recurrence obtained by conditioning on the random arrival edge and position. To make this recurrence computationally effective, we exploit dihedral symmetry of the regular polygon and show that $V(S)$ is invariant under rotations and reflections, enabling canonicalization and symmetry-reduced dynamic programming. For small $n$, we evaluate the recurrence accurately using deterministic numerical integration over piecewise-linear distance regions,; for larger $n$, we estimate the expected total cost via direct Monte Carlo simulation of the online process and report $95%$ confidence intervals. Our computations validate the recurrence (including a closed-form check for the square, $n=4$) and indicate that the total expected cost increases with $n$, while the per-customer expected travel distance grows gradually as remaining free vertices become farther on average. \keywords{Online algorithms \and Facility assignment \and Expected cost \and Regular polygons \and Symmetry reduction \and Monte Carlo}

💡 Research Summary

The paper investigates the expected total travel distance incurred by a greedy online facility assignment algorithm on a regular n‑gon. Each vertex of the polygon hosts a unit‑capacity facility, and customers arrive one by one. An arrival first selects an edge uniformly at random and then a point uniformly along that edge. The distance from the arrival point to any vertex is measured by the shortest walk along the polygon edges (the cycle graph metric). Upon arrival, the algorithm irrevocably assigns the customer to the nearest free facility; if several facilities are equally near, a uniform random tie‑break is performed.

The authors introduce a state‑dependent value function V(S), where S⊆V denotes the set of already occupied vertices. V(S) is defined as the expected remaining total cost from state S until all facilities are filled. By conditioning on the random edge and the position t∈