Kernelization of Discrete Optimization Problems on Parallel Architectures

TL;DR

This paper develops and evaluates parallel algorithms for Kernelization in discrete optimization, specifically for a colored K-clique problem, demonstrating over 50% efficiency improvements and smaller kernels.

Contribution

It introduces parallel implementations of Kernelization algorithms for a specific discrete optimization problem, improving speed and kernel size reduction.

Findings

Parallel algorithms achieve over 50% efficiency improvement.

Parallel methods produce smaller kernels.

Evaluation focused on a colored K-clique problem.

Abstract

There are existing standard solvers for tackling discrete optimization problems. However, in practice, it is uncommon to apply them directly to the large input space typical of this class of problems. Rather, the input is preprocessed to look for simplifications and to extract the core subset of the problem space, which is called the Kernel. This pre-processing procedure is known in the context of parameterized complexity theory as Kernelization. In this thesis, I implement parallel versions of some Kernelization algorithms and evaluate their performance. The performance of Kernelization algorithms is measured either by the size of the output Kernel or by the time it takes to compute the kernel. Sometimes the Kernel is the same as the original input, so it is desirable to know this, as soon as possible. The problem scope is limited to a particular type of discrete optimisation problem which is a version of the K-clique problem in which nodes of the given graph are pre-coloured legally using k colours. The final evaluation shows that my parallel implementations achieve over 50% improvement in efficiency for at least one of these algorithms. This is attained not just in terms of speed, but it is also able to produce a smaller kernel.