Experimental Algorithm for the Maximum Independent Set Problem

TL;DR

This paper presents an experimental algorithm for exactly solving the maximum independent set problem in graphs, utilizing a novel approach based on poset partition techniques and tested on random graphs.

Contribution

It introduces a new algorithm combining poset partition methods with graph theory for maximum independent set, with proven correctness and an estimated polynomial running time.

Findings

Algorithm correctly finds maximum independent sets in tested graphs.

Theoretical running time estimated at O(n^8).

Algorithm validated through experiments on random graphs.

Abstract

We develop an experimental algorithm for the exact solving of the maximum independent set problem. The algorithm consecutively finds the maximal independent sets of vertices in an arbitrary undirected graph such that the next such set contains more elements than the preceding one. For this purpose, we use a technique, developed by Ford and Fulkerson for the finite partially ordered sets, in particular, their method for partition of a poset into the minimum number of chains with finding the maximum antichain. In the process of solving, a special digraph is constructed, and a conjecture is formulated concerning properties of such digraph. This allows to offer of the solution algorithm. Its theoretical estimation of running time equals to is $O(n^{8})$, where $n$ is the number of graph vertices. The offered algorithm was tested by a program on random graphs. The testing the confirms correctness of the algorithm.