Marijn Heule
Publications
Doubly Saturated Ramsey Graphs: A Case Study in Computer-Assisted Mathematical Discovery
Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to generate and formalize correctness proofs in Lean. This case study highlights the potential of integrating automated reasoning, large language models, and formal verification to accelerate mathematical discovery. We argue that such tool-driven workflows will play an increasingly central role in experimental mathematics.
Orbitopal Fixing in SAT
Despite their sophisticated heuristics, boolean satisfiability (SAT) solvers are still vulnerable to symmetry, causing them to visit search regions that are symmetric to ones already explored. While symmetry handling is routine in other solving paradigms, integrating it into state-of-the-art proof-producing SAT solvers is difficult: added reasoning must be fast, non-interfering with solver heuristics, and compatible with formal proof logging. To address these issues, we present a practical static symmetry breaking approach based on orbitopal fixing, a technique adapted from mixed-integer programming. Our approach adds only unit clauses, which minimizes downstream slowdowns, and it emits succinct proof certificates in the substitution redundancy proof system. Implemented in the satsuma tool, our methods deliver consistent speedups on symmetry-rich benchmarks with negligible regressions elsewhere.