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种自然的动态变体，其中需要动态维护一个将 $n$ 个元素划分为两个大小为 $n/2$ 的簇的划分。在运行过程中，在线算法会收到一系列请求，每个请求是一对元素：簇间请求花费一个单位成本，而簇内请求则免费。算法可以更改划分，每个更改簇的元素支付一个单位成本。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 等人，DISC 2016]。尽管自原始工作以来，已有若干显著改进的结果，但它们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，即使允许使用随机化，Avin 等人的算法仍实现了目前已知的最佳竞争比。在本文中，我们提出了首个随机在线算法，该算法突破了这一自然的二次障碍，在没有资源增强且针对任意请求序列的情况下，实现了 $\tilde{O}(n^{23/12})$ 的竞争比。