Evolving Local Corrections for Global Constructions in Combinatorics

TL;DR

This paper explores how iterative local corrections can lead to global solutions in combinatorics, using computational experiments to identify structural patterns and propose conjectures for complex problems.

Contribution

It introduces a computational framework for applying and refining local correction steps in combinatorial problems, providing new algorithms and structural insights.

Findings

Identified potential correction rules for graph reconstruction

Developed sign-reversing involutions for Latin square parity

Suggested local exchange policies for Rota's Basis Conjecture

Abstract

Many open problems in combinatorics admit reformulations in which a global construction can be achieved by the repeated application of small, finite correcting steps. This paper presents three computational case studies of this principle, carried out using AlphaEvolve as an experimental engine for proposing and iteratively refining such certificates. The problems we studied are: reconstruction of bipartite and planar graphs from vertex-deleted subgraphs; the Alon-Tarsi parity problem for Latin squares, approached via sign-reversing involutions built from local trades; Rota's Basis Conjecture, studied through local exchange policies on collections of bases. In these three problems the correcting steps take the form of a reconstruction rule, a parity-reversing involution, and a transversal family of bases, respectively. For each problem, we describe the experimental setup, the scoring protocols, and the outcomes of the searches, leading to concrete conjectures concerning the existence and structure of the relevant correcting steps. The aim is not to claim proofs, but rather to produce explicit algorithms and to reveal structural patterns that appear amenable to subsequent analysis by traditional mathematical methods.