Maximum independent sets of the 120-cell and other regular polytopes

Abstract

A d-code in a graph is a set of vertices such that all pairwise distances are at least d. As part of a study of d-codes of three-and four-dimensional regular polytopes, the maximum independent set order of the 120-cell is calculated. A linear program based on counting arguments leads to an upper bound of 221. An independent set of order 110 in the antipodal collapse of the 120-cell (also known as the hemi-120-cell) gives a lower bound of 220 for the 120-cell itself. The gap is closed by the computation described here, with the result that the maximum independent set order of the 120-cell is 220. All maximum d-code orders of the icosahedron, dodecahedron, 24-cell, 600-cell and 120-cell are reported.

On scalable parallel recursive backtracking

“…To the best of our knowledge, this paper is the first to experimentally solve it; more information on this instance can be found in [23]. Lastly, the 60-cell graph is a 4-regular graph (every vertex has exactly 4 neighbors) with applications in chemistry [16]. Prior to this work, we solved the 60-cell using a serial algorithm which ran for almost a full week [17].…”

“…To the best of our knowledge, the most efficient existing parallel algorithms that solve problems similar to those we consider were only able to scale to less than a few thousand (or only a few hundred) cores [10], [11], [15]. One of our main motivations was to solve extremely hard instances of the VERTEX COVER problem such as the 60-cell graph [16]. In earlier work, we first attempted to tackle the problem by improving the efficiency of our serial algorithm [17].…”

The number of independent sets in a connected graph and its complement

“…In [17], S. Wagner and I. Gutman wrote a survey of results and techniques on the Hosoya index and Merrifield-Simmons index. Other recent results on the number of independent sets can be found in [2,4,3,9].…”

A decentralized load balancing strategy for parallel search-three optimization. (c2010)

“…A notorious example is the partial model of the 120-cell graph which is a 4-regular graph (i.e. every vertex has degree 4) on 300 vertices and 600 edges [14]. The significance of exact algorithms and the need for accurate solutions has greatly increased in the last five decades.…”