K-way Balanced Graph Partitioning for Parallel Computing

In modern computing, high-performance computing (HPC) and parallel computing require most of the decision-making in terms of distributing the payloads (input) uniformly across the available set of resources, majorly processors; the former deals with the hardware and its better utilization. In parallel computing, a larger, complex problem is broken down into multiple smaller calculations and executed simultaneously on several processors. The efficient use of resources (processors) plays a vital role in achieving the maximum throughput which necessitates uniform load distribution across available processors, i.e. load balancing. The load balancing in parallel computing is modeled as a graph partitioning problem. In the graph partitioning problem, the weighted nodes represent the computing cost at each node, and the weighted edges represent the communication cost between the connected nodes. The goal is to partition the graph G into k partitions such that: I) the sum of weights on the nodes is approximately equal for each partition, and, II) the sum of weights on the edges across different partitions is minimum. In this paper, a novel node-weighted and edge-weighted k-way balanced graph partitioning (NWEWBGP) algorithm of O(n x n) is proposed. The algorithm works for all relevant values of k, meets or improves on earlier algorithms in terms of balanced partitioning and lowest edge-cut. For evaluation and validation, the outcome is compared with the ground truth benchmarks.

Simplex Transformations and the Multiway Cut Problem

We consider multiway cut, a basic graph partitioning problem in which the goal is to find the minimum weight collection of edges disconnecting a given set of special vertices called terminals. Multiway cut admits a well-known simplex embedding relaxation, where rounding this embedding is equivalent to partitioning the simplex. Current best-known solutions to the problem are comprised of a mix of several different ingredients, resulting in intricate algorithms. Moreover, the best of these algorithms is too complex to fully analyze analytically, and a computer was partly used in verifying its approximation factor. We propose a new approach to simplex partitioning and the multiway cut problem based on general transformations of the simplex that allow dependencies between the different variables. Our approach admits much simpler algorithms and, in addition, yields an approximation guarantee for the multiway cut problem that (roughly) matches the current best computer-verified approximation factor.

Computational Comparison of Major Proposed Methods for Graph Partitioning Problem

k-way graph partitioning is an NP-complete problem, which is applied to various tasks such as route planning, image segmentation, community detection, and high-performance computing. The approximate methods constitute a useful solution for these types of problems. Thus, many research studies have focused on developing meta-heuristic algorithms to tackle the graph partitioning problem. Local search is one of the earliest methods that has been applied efficiently to this type of problem. Recent studies have explored various types of local search methods and have improved them such that they can be used with the partitioning process. Moreover, local search methods are widely integrated with population-based approaches, to provide the best diversification and intensification for the problem space. This study emphasizes the local search approaches, as well as their combination with other graph partitioning approaches. At present, none of the surveys in the literature has focused on this class of state of the art approaches in much detail. In this study, the vital parts of these approaches including neighborhood structure, acceptance criterion, and the ways of combining them with other approaches, are highlighted. Additionally, we provide an experimental comparison that shows the variance in the performance of the reviewed methods. Hence, this study clarifies these methods to show their advantages and limitations for the targeted problem, and thus can aid in the direction of research flow towards the area of graph partitioning.