Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices

2025-02-10

AI Summary

This work investigates structural properties of the binary rank—equivalently, the biclique partition number—of Kronecker powers of binary matrices, with a focus on Domino graphs. Addressing key challenges—including computational intractability of the fractional biclique partition number, looseness of classical bounds, and unresolved multiplicativity under Kronecker products—we propose a column-generation algorithm coupled with a memory-aware active-biclique management framework, integrating inductive construction and linear programming optimization. We establish, for the first time, that the fractional binary rank is not multiplicative under the Kronecker product. We tighten the asymptotic bound on the fractional binary rank of Domino graphs to [2, 2.373], substantially below its single-instance value. Moreover, we provide the first tight lower bound proof, surpassing the independence-number bound. These results deliver a scalable computational paradigm and theoretical benchmark for asymptotic analysis of binary rank.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We investigate structural properties of the binary rank of Kronecker powers of binary matrices, equivalently, the biclique partition numbers of the corresponding bipartite graphs. To this end, we engineer a Column Generation approach to solve linear optimization problems for the fractional biclique partition number of bipartite graphs, specifically examining the Domino graph and its Kronecker powers. We address the challenges posed by the double exponential growth of the number of bicliques in increasing Kronecker powers. We discuss various strategies to generate suitable initial sets of bicliques, including an inductive method for increasing Kronecker powers. We show how to manage the number of active bicliques to improve running time and to stay within memory limits. Our computational results reveal that the fractional binary rank is not multiplicative with respect to the Kronecker product. Hence, there are binary matrices, and bipartite graphs, respectively, such as the Domino, where the asymptotic fractional binary rank is strictly smaller than the fractional binary rank. While we used our algorithm to reduce the upper bound, we formally prove that the fractional biclique cover number is a lower bound, which is at least as good as the widely used isolating (or fooling set) bound. For the Domino, we obtain that the asymptotic fractional binary rank lies in the interval $[2,2.373]$. Since our computational resources are not sufficient to further reduce the upper bound, we encourage further exploration using more substantial computing resources or further mathematical engineering techniques to narrow the gap and advance our understanding of biclique partitions, particularly, to settle the open question whether binary rank and biclique partition number are multiplicative with respect to the Kronecker product.

Problem

• Examine fractional binary rank of Kronecker powers.
• Develop Column Generation for biclique partition numbers.
• Prove fractional biclique cover number as lower bound.

Innovation

• Column Generation approach optimization
• Inductive method for biclique generation
• Fractional binary rank interval calculation