On $\varepsilon$-Matrix Product Factorization of graphs

2026-07-29

AI Summary

This work addresses the infeasibility of exact matrix product factorization for graphs due to rigid structural constraints by introducing an $\varepsilon$-approximate decomposition framework. This framework permits the product of adjacency matrices of factor graphs to differ from the original graph in at most $\varepsilon n^2$ entries while preserving the two-step witness structure of edges. We establish, for the first time, equivalent matrix and witness characterizations for approximate graph matrix decompositions, revealing that sparse errors effectively circumvent the rigidity inherent in exact factorizations. By integrating tools from graph theory, matrix analysis, and combinatorial construction—particularly neighborhood Cartesian products and path-counting constraints—we demonstrate that complete graphs, expander graphs, and trees all admit efficient $\varepsilon$-approximate decompositions with $\varepsilon = O(1/n)$, substantially relaxing the number-theoretic conditions traditionally required for exact decompositions.

Abstract

We introduce an approximate version of matrix product factorization for graphs. A simple graph $G$ on $n$ vertices is said to admit an $\varepsilon$-matrix product factorization if there exist simple graphs $H$ and $K$ on the same vertex set such that $A(H)A(K)$ and $A(G)$ disagree in at most $\varepsilon n^{2}$ entries. This Hamming-type relaxation preserves, outside the error set, the exact interpretation of each edge as having a unique $H$-then-$K$ two-step witness. We establish equivalent matrix, and witness formulations, showing that the sets $N_H(w)\times N_K(w)$ form an approximate disjoint decomposition of the ordered adjacency relation of $G$, and we derive quantitative constraints involving walk counts and the degrees of the factor graphs. We then construct approximate factorizations for several graph families. Every complete graph $K_n$ has matrix-product-factorization distance $O(1/n)$, despite the exact congruence obstruction that permits exact factorization only when $n\equiv 1\pmod 4$. More generally, a blow-up of a fixed graph on $r$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le r/n$, and the construction is exact whenever every non-isolated part has even order. For bipartite graphs, we give one-sided factorizations that realize one orientation of almost all edges. In particular, every tree on $n\ge2$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le 1/n$, although no nontrivial tree is exactly factorizable. These results show that rigid exact obstructions may disappear under a vanishing proportion of entrywise errors.

Problem

• matrix product factorization
• graph decomposition
• approximate factorization
• adjacency matrix
• Hamming error

Innovation

• epsilon-matrix product factorization
• graph decomposition
• approximate factorization
• witness paths
• Hamming-type relaxation