In the online bisection problem one has to 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 a first randomized online algorithm that breaks this natural barrier and achieves a competitive ratio of $\tilde{O}(n^{27/14})$ without resource augmentation and for an arbitrary sequence of requests.

翻译：在在线二分问题中，需要维护一个将$n$个元素划分为两个基数均为$n/2$的簇的划分。在运行过程中，在线算法会接收到一系列请求，每个请求为一对元素：跨簇请求产生一单位代价，而簇内请求则免费。算法可调整划分，每次更改元素所属簇需支付一单位代价。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，要么假设某种形式的资源增强（如更大的簇）。此外，即便允许随机化，Avin等人的算法仍达到已知最佳竞争比。本文提出首个无资源增强、针对任意请求序列的随机在线算法，其竞争比突破该自然界限，达到$\tilde{O}(n^{27/14})$。