The online bisection problem is a natural dynamic variant of the classic optimization problem, where one has to dynamically maintain a partition of $n$ elements into two clusters of cardinality $n/2$. During runtime, an online algorithm is given a sequence of requests, each being a pair of elements: an inter-cluster request costs one unit while an intra-cluster one is free. The algorithm may change the partition, paying a unit cost for each element that changes its cluster. This natural problem admits a simple deterministic $O(n^2)$-competitive algorithm [Avin et al., DISC 2016]. While several significant improvements over this result have been obtained since the original work, all of them either limit the generality of the input or assume some form of resource augmentation (e.g., larger clusters). Moreover, the algorithm of Avin et al. achieves the best known competitive ratio even if randomization is allowed. In this paper, we present the first randomized online algorithm that breaks this natural quadratic barrier and achieves a competitive ratio of $\tilde{O}(n^{23/12})$ without resource augmentation and for an arbitrary sequence of requests.

翻译：在线二分问题是经典优化问题的一个自然动态变体，在该问题中需要动态地将$n$个元素划分为两个大小为$n/2$的簇。运行期间，在线算法接收一系列请求，每个请求为一对元素：跨簇请求产生一单位成本，而簇内请求则免费。算法可改变划分，每个改变所属簇的元素需支付一单位成本。这一自然问题存在一个简单的确定性$O(n^2)$-竞争比算法[Avin等，DISC 2016]。尽管自原始工作以来已取得多项显著改进，但这些改进要么限制了输入的通用性，要么假设了某种形式的资源增强（例如更大的簇）。此外，即便允许随机化，Avin等人的算法仍能达到已知最佳竞争比。本文首次提出一种随机化在线算法，无需资源增强且适用于任意请求序列，突破了这一自然的二次瓶颈，实现了$\tilde{O}(n^{23/12})$的竞争比。