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

TL;DR

This work analyzes Online Facility Assignment on discrete grid graphs under hard capacities, revealing geometry-induced failure modes for natural local rules: a capacity-aware CS-Voronoi heuristic can suffer Zone-Collapse and Boundary-Oscillation, while nearest-available Greedy can incur costly oscillations. It introduces a batching-plus-min-cost-flow remediation (BMCF) to provide bounded lookahead, but proves that cross-batch coordination is not captured by per-batch optimizations, illustrating a batch-boundary trap that yields large cost gaps relative to OPT in adversarial constructions. The paper then proposes practical mitigations—capacity reservation, scarcity-penalized batch costs, and staggered batching triggers—and shows empirically that these guardrails reduce tail risk while preserving reasonable performance on benign inputs. The results highlight the importance of guardrails for grid metrics with capacity constraints and outline promising directions for principled semi-online analysis and parameter tuning in realistic workloads.

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.

Figures (5)

• Figure 1: Zone-Collapse: once a central facility depletes, a large nearby region loses its local option and subsequent requests incur a sharp distance increase.
• Figure 2: Boundary-Oscillation: small capacity changes can flip the minimizing facility across a wide staircase bisector, enabling repeated costly redirections.
• Figure 3: Illustration of capacity-sensitive scoring and how discrete capacity changes can alter the minimizing facility.
• Figure 4: Oscillation trap for Randomized Greedy under hard capacities. A first request $R_1$ arrives at a location equidistant from two unit-capacity facilities $F(L)$ and $F(R)$. Randomized Greedy breaks the tie by a coin flip and fills one facility. An adversary then releases a second request $R_2$ at the location of the filled facility, forcing assignment to the other facility and incurring the full cross-distance.
• Figure 5: Batch-overconcentration failure mode for BMCF. Batch 1 is assigned locally (minimal local cost) and saturates the nearest facility; Batch 2 then arrives nearby but must traverse distance $\Delta$ to the only remaining capacity.

Theorems & Definitions (4)

• proposition thmcounterproposition: Best-Case Behavior
• proof
• lemma thmcounterlemma: BMCF suffers a batch-boundary trap with $\Omega(\Delta)$ blow-up
• proof