Load Balancing under Adaptive Bin Deletions

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.