[Paper Review] Dynamic Balanced Graph Partitioning

Chen Avin, Marcin Bieńkowski+3 more|arXiv (Cornell University)|Nov 6, 2015

Optimization and Search Problems46 references4 citations

TL;DR

This paper introduces the online dynamic balanced graph partitioning problem, where nodes are partitioned into clusters of equal size to minimize inter-cluster communication and migration costs under time-varying communication patterns. It presents an $O(k\log k)$-competitive deterministic algorithm for the general case with constant cluster-size augmentation, and proves a lower bound of $k$ on the competitive ratio for deterministic algorithms even with augmentation.

ABSTRACT

This paper initiates the study of the classic balanced graph partitioning problem from an online perspective: Given an arbitrary sequence of pairwise communication requests between $n$ nodes, with patterns that may change over time, the objective is to service these requests efficiently by partitioning the nodes into $\ell$ clusters, each of size $k$, such that frequently communicating nodes are located in the same cluster. The partitioning can be updated dynamically by migrating nodes between clusters. The goal is to devise online algorithms which jointly minimize the amount of inter-cluster communication and migration cost. The problem features interesting connections to other well-known online problems. For example, scenarios with $\ell=2$ generalize online paging, and scenarios with $k=2$ constitute a novel online variant of maximum matching. We present several lower bounds and algorithms for settings both with and without cluster-size augmentation. In particular, we prove that any deterministic online algorithm has a competitive ratio of at least $k$, even with significant augmentation. Our main algorithmic contributions are an $O(k \log{k})$-competitive deterministic algorithm for the general setting with constant augmentation, and a constant competitive algorithm for the maximum matching variant.

Motivation & Objective

• To address the challenge of dynamically repartitioning nodes in response to time-varying communication patterns in distributed systems.
• To jointly minimize inter-cluster communication and node migration costs in an online setting with no prior knowledge of future requests.
• To establish theoretical bounds on the performance of online algorithms for balanced graph partitioning, especially under constraints on cluster size.
• To explore connections to classic online problems such as online paging and online matching.
• To develop competitive online algorithms for dynamic clustering in virtualized environments like cloud computing.

Proposed method

• Models the problem as an online balanced repartitioning (BRP) task where nodes are dynamically migrated between clusters of fixed size $k$ to minimize communication and migration costs.
• Uses competitive analysis to compare online algorithms against an optimal offline algorithm with full knowledge of the request sequence.
• Employs an averaging argument over multiple offline algorithms to derive lower bounds on the competitive ratio.
• Designs a deterministic algorithm that achieves $O(k\log k)$-competitiveness using cluster-size augmentation.
• Applies a technique of partitioning nodes into $k$ cyclic cuts $\textsc{cut}(j)$ to explore alternative clusterings and bound performance.
• Proves lower bounds via adversarial request sequences that force online algorithms into suboptimal configurations, showing inherent limitations of deterministic approaches.

Experimental results

Research questions

• RQ1 What is the best possible competitive ratio achievable by a deterministic online algorithm for dynamic balanced graph partitioning without cluster-size augmentation?
• RQ2 Can a constant-competitive algorithm be achieved for the special case where $k=2$, corresponding to an online variant of maximum matching?
• RQ3 How does cluster-size augmentation affect the competitive ratio, and can it enable better algorithmic guarantees?
• RQ4 Is it possible to design an online algorithm whose competitive ratio is independent of the number of clusters $\ell$?
• RQ5 What are the fundamental limits of online dynamic partitioning when communication patterns evolve over time?

Key findings

• Any deterministic online algorithm for dynamic balanced graph partitioning has a competitive ratio of at least $k$, even with significant cluster-size augmentation.
• An $O(k\log k)$-competitive deterministic algorithm is presented for the general case when a constant amount of cluster-size augmentation is allowed.
• For the special case with $k=2$, the paper proves a lower bound of 3 on the competitive ratio for deterministic online algorithms.
• The paper establishes that the $O(k\log k)$ competitive ratio is tight up to a constant factor under constant augmentation.
• The competitive ratio of $k$ is proven to be unavoidable even with $O(1)$-augmentation, indicating that $k$-dependence is inherent in the problem.
• The results show that the offline optimal solution can be approximated within a factor of $O(k\log k)$ using constant augmentation, which is a significant improvement over the trivial $O(k^2 \ell^2)$ bound.