A Subquadratic Bound for Online Bisection
Marcin Bienkowski, Stefan Schmid
TL;DR
This work addresses online bisection, the dynamic problem of maintaining a partition of $n$ elements into two equal-sized clusters while minimizing inter-cluster requests and migration costs. It introduces Icb, a randomized, component-preserving algorithm that splits epochs into a deterministic first stage guided by a gcd-based rebalancing strategy and a second randomized stage that selects partitions uniformly from the remaining feasible set, effectively reducing the number of elements moved. The main result is a subquadratic competitive ratio of $O\big(n^{23/12}\sqrt{\log n}\big)$ without resource augmentation for arbitrary request sequences, advancing beyond the long-standing $O(n^2)$ barrier. The approach combines number-theoretic structure with online rebalancing analysis and ties the second stage to metrical task systems, offering new techniques for online balanced partitioning with practical implications for data-center VM consolidation and network optimization.
Abstract
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.