On an Algorithm for Isomorphism-Free Generations of Combinatorial Objects

TL;DR

This paper introduces an efficient algorithm using bitwise operations to generate semi-canonical and canonical binary matrices with fixed row and column sums, avoiding exhaustive enumeration.

Contribution

It presents a novel algorithm for isomorphism-free generation of combinatorial objects, specifically binary matrices with fixed row and column sums, using semi-canonical and canonical forms.

Findings

Algorithm efficiently finds one representative per equivalence class.

Uses bitwise operations to simplify the search process.

Reduces computational complexity compared to brute-force methods.

Abstract

In the work are defined the concepts semi-canonical and canonical binary matrix. What is described is an algorithm solving the combinatorial problem for finding the semi-canonical matrices in the set \Lambda_n^k consisting of all n\times n binary matrices having exactly k 1's in every row and every column without perambulating all elements. In the described algorithm bitwise operations are substantially used. In this way it becomes easier to find the solution to the problem for receiving one representative from every equivalence class regarding the introduced in the article equivalence relation in the set \Lambda_n^k . The last problem is equivalent to the problem for finding all canonical matrices in \Lambda_n^k .