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，要求动态维护$n$个元素的两个基数均为$n/2$的簇划分。运行时，在线算法接收一系列请求，每个请求由一对元素组成：跨簇请求成本为一个单位，而簇内请求则为零成本。算法可更改划分，每个改变所属簇的元素需支付一单位成本。该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取得若干显著改进，但这些改进要么限制了输入的通用性，要么假设了某种资源增强（例如更大的簇）。此外，即使允许随机化，Avin等人的算法仍实现了已知最优竞争比。在本文中，我们提出了首个无需资源增强且针对任意请求序列、竞争比为$\tilde{O}(n^{23/12})$的随机化在线算法，打破了这一自然的二次界限。