Load Balancing under Adaptive Bin Deletions

2026-07-07

AI Summary

This work addresses the problem of efficiently reallocating tasks (balls) upon the adversarial deletion of bins in a dynamic setting, where an adaptive adversary removes one bin per round and the algorithm must reassign its balls to maintain load balance while controlling total recourse cost. The paper establishes, for the first time, that uniform random reallocation achieves both linear total recourse and a near-optimal maximum load of $O(\log \log n)$ against an adaptive adversary. Furthermore, it introduces a novel two-group reallocation strategy for balls from the deleted bin, surpassing the limitations of conventional single-group approaches: with only two choices ($d=2$), this method simultaneously guarantees linear recourse and efficient load balancing.

Abstract

We analyze a balls-and-bins game against an adaptive adversary that sequentially deletes bins. Starting with $n$ balls distributed across $n$ bins, the adversary deletes a bin in each step, forcing the algorithm to redistribute its balls to surviving bins. We prove that after $n/2$ rounds, uniform random redistribution yields optimal $O(n)$ recourse and $O(\frac{\log n}{\log \log n})$ maximum load. Furthermore, we show that applying the ``power of two choices'' reduces the maximum load to $O(\log \log n)$ while maintaining linear recourse. We also consider a variation of this game where the balls from the deleted bin are partitioned evenly among $d \ll n$ random bins rather than being redistributed independently. We demonstrate that keeping the balls together ($d=1$), which gives small maximum load and recourse against an oblivious adversary, fails against an adaptive adversary. Nevertheless, we show that splitting the balls into just two groups ($d=2$) is sufficient to recover linear recourse and efficient load balancing in the adaptive setting.

Problem

• load balancing
• adaptive adversary
• balls-and-bins
• bin deletion
• maximum load

Innovation

• adaptive adversary
• balls-and-bins
• power of two choices
• load balancing
• recourse