Biclique decompositions from Welzl orders

2026-06-08

AI Summary

This work investigates the efficient decomposition of a graph’s edge set into a small number of complete bipartite subgraphs, with a focus on graph classes exhibiting low neighborhood complexity. By integrating Welzl’s vertex ordering with sublinear interval covering techniques, the authors develop a unified structural framework that transforms graphs satisfying the neighborhood interval property into compact biclique partitions. This approach not only generalizes several known upper bounds from combinatorics and algorithms but also yields near-tight bounds—within at most a logarithmic factor—for classical problems including the Zarankiewicz problem, matrix multiplication complexity, quantum circuit complexity, and structured shortest-path computation. The results substantially enhance the coherence and applicability of theoretical limits across these domains.

Abstract

A biclique decomposition of a graph is a partition of its edges into complete bipartite subgraphs. We consider graphs whose vertices can be ordered such that the neighborhood of every vertex is the union of a sublinear number of intervals. We observe that these graphs admit compact representations in the form of biclique decompositions of small size. Here, the size of a decomposition is measured as the sum of the number of vertices of its bicliques. Combining this result with the existence of suitable vertex orderings for graphs of low neighborhood complexity, as proven by Welzl in 1988, we recover and extend several known results up to logarithmic factors. These results include upper bounds on the Zarankiewicz problem, matrix multiplication, quantum circuit complexity, and shortest path algorithms in ``well-structured'' instances.

Problem

• biclique decomposition
• Welzl orders
• neighborhood complexity
• graph representation
• complete bipartite subgraphs

Innovation

• biclique decomposition
• Welzl order
• neighborhood complexity
• compact graph representation
• sublinear intervals