Design by Measure and Conquer, A Faster Exact Algorithm for Dominating Set

TL;DR

This paper introduces a novel measure and conquer-based methodology for designing faster exact algorithms for the Dominating Set problem, achieving new state-of-the-art exponential time bounds through computer-aided analysis and reduction rules.

Contribution

It presents a new approach that integrates measure and conquer into the algorithm design process, leading to improved exponential algorithms for Dominating Set.

Findings

Developed algorithms with $O(1.5134^n)$ and $O(1.5063^n)$ time complexity.

Used quasiconvex programming to analyze and optimize algorithms.

Achieved the fastest known exact algorithms for Dominating Set.

Abstract

The measure and conquer approach has proven to be a powerful tool to analyse exact algorithms for combinatorial problems, like Dominating Set and Independent Set. In this paper, we propose to use measure and conquer also as a tool in the design of algorithms. In an iterative process, we can obtain a series of branch and reduce algorithms. A mathematical analysis of an algorithm in the series with measure and conquer results in a quasiconvex programming problem. The solution by computer to this problem not only gives a bound on the running time, but also can give a new reduction rule, thus giving a new, possibly faster algorithm. This makes design by measure and conquer a form of computer aided algorithm design. When we apply the methodology to a Set Cover modelling of the Dominating Set problem, we obtain the currently fastest known exact algorithms for Dominating Set: an algorithm that uses $O(1.5134^n)$ time and polynomial space, and an algorithm that uses $O(1.5063^n)$ time.