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arXiv · 2602.05953

Competitive Analysis of Online Facility Assignment Algorithms on Discrete Grid Graphs: Performance Bounds and Remediation Strategies

Lamya Alif·Raian Tasnim Saoda·Sumaiya Afrin·Md. Rawha Siddiqi Riad·Md. Tanzeem Rahat·Md Manzurul Hasan
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Abstract

We study the \emph{Online Facility Assignment} (OFA) problem on a discrete $r\times c$ grid graph under the standard model of Ahmed, Rahman, and Kobourov: a fixed set of facilities is given, each with limited capacity, and an online sequence of unit-demand requests must be irrevocably assigned upon arrival to an available facility, incurring Manhattan ($L_1$) distance cost. We investigate how the discrete geometry of grids interacts with capacity depletion by analyzing two natural baselines and one capacity-aware heuristic. First, we give explicit adversarial sequences on grid instances showing that purely local rules can be forced into large competitive ratios: (i) a capacity-sensitive weighted-Voronoi heuristic (\textsc{CS-Voronoi}) can suffer cascading \emph{region-collapse} effects when nearby capacity is exhausted; and (ii) nearest-available \textsc{Greedy} (with randomized tie-breaking) can be driven into repeated long reassignments via an \emph{oscillation} construction. These results formalize geometric failure modes that are specific to discrete $L_1$ metrics with hard capacities. Motivated by these lower bounds, we then discuss a semi-online extension in which the algorithm may delay assignment for up to $τ$ time steps and solve each batch optimally via a min-cost flow computation. We present this batching framework as a remediation strategy and delineate the parameters that govern its performance, while leaving sharp competitive guarantees for this semi-online variant as an open direction.

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Lamya Alif, Raian Tasnim Saoda, Sumaiya Afrin, Md. Rawha Siddiqi Riad, Md. Tanzeem Rahat, Md Manzurul Hasan. 2026-02-05. Competitive Analysis of Online Facility Assignment Algorithms on Discrete Grid Graphs: Performance Bounds and Remediation Strategies. https://arxiv.org/abs/2602.05953

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