Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer (1605.00723v1)

Abstract: The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set N = ${1, 2, ...}$ of natural numbers be divided into two parts, such that no part contains a triple $(a,b,c)$ with $a^2 + b^2 = c^2$ ? A prize for the solution was offered by Ronald Graham over two decades ago. We solve this problem, proving in fact the impossibility, by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers. An important role is played by dedicated look-ahead heuristics, which indeed allowed to solve the problem on a cluster with 800 cores in about 2 days. Due to the general interest in this mathematical problem, our result requires a formal proof. Exploiting recent progress in unsatisfiability proofs of SAT solvers, we produced and verified a proof in the DRAT format, which is almost 200 terabytes in size. From this we extracted and made available a compressed certificate of 68 gigabytes, that allows anyone to reconstruct the DRAT proof for checking.

• The paper demonstrates a breakthrough by using the Cube-and-Conquer method to prove that numbers up to 7825 cannot be partitioned without forming a Pythagorean triple.
• The approach integrates look-ahead strategies with conflict-driven clause learning, effectively splitting the problem into billions of manageable subproblems.
• The results, verified by a DRAT proof and consuming roughly 35,000 CPU hours, showcase the potential of SAT solvers in resolving complex combinatorial challenges.

Analysis of the Boolean Pythagorean Triples Problem using the Cube-and-Conquer Approach

The paper "Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer," authored by Marijn J. H. Heule, Oliver KuLLMann, and Victor W. Marek, addresses an enduring problem in Ramsey Theory concerning the partitioning of natural numbers into two sets, avoiding a configuration of Pythagorean triples. Utilizing the hybrid SAT-solving method known as Cube-and-Conquer (C&C), the authors provide a computational solution to this problem, thereby establishing the impossibility of such a partition for the set of numbers up to 7825.

The Boolean Pythagorean Triples (BPT) problem is rooted in extremal combinatorics, exploring the possibility that the infinite set $N = \{1,2,\dots\}$ can be divided such that no Pythagorean triple $(a,b,c)$ with $a^2 + b^2 = c^2$ exists within either subset. The resolution of this problem by the authors demonstrates an innovative application of SAT-solving techniques to mathematical problems traditionally explored through different computational means.

Methodology

The authors leverage the Cube-and-Conquer SAT paradigm, a potent synthesis of look-ahead and conflict-driven clause learning (CDCL) solvers. The methodology divides the problem into numerous subproblems (cubes), solved independently, exploiting parallel computing capabilities. The C&C approach is especially suited for intractable combinatorial problems, enabling the partition of these problems into billions of smaller, tractable subproblems.

Their results rest heavily on advanced heuristics for partitioning (or "splitting") of the SAT instances, making efficient use of available computational resources. The splitting heuristics are crucial, leveraging look-ahead mechanics to guide the partitioning process. This ensures computationally feasible subproblems while optimizing solver performance by focusing on the most impactful bifurcation decisions.

Results and Implications

The authors' computational effort required roughly 35,000 CPU hours over a cluster environment, demonstrating the feasibility but also the computational intensity of the task. The conclusion that no partition can prevent a Pythagorean triple within the set $\{1, \dots, 7825\}$ is robust, supported by a verified DRAT proof, with the process generating an unsatisfiability proof of nearly 200 terabytes, although compressed to 68 gigabytes for practical validation.

The resolution of the BPT problem provides significant insights into both the combinatorial structures of Ramsey Theory and the application of SAT solvers to previously unresolved theoretical problems. The computational approach underscores the potential for SAT-oriented techniques to address complex mathematical inquiries traditionally limited to more classical theorem-proving methodologies.

Theoretical and Practical Considerations

Theoretically, the approach challenges existing paradigms concerning the tractability of certain Ramsey-type problems, providing a computational perspective where traditional combinatorial approaches face limitations. Practically, as high-performance computing capabilities advance, this method offers a prototype for addressing similarly complex problems, expanding the toolkit available to researchers in extremal combinatorics and beyond.

Future advancements could focus on the development of more efficient heuristics and algorithmic optimizations within the C&C framework, targeted at improving solver performance and scalability further. The question remains whether inherently combinatorial problems like the BPT problem cultivate computational boundaries that even advanced SAT solvers may struggle to transcend without corresponding improvements in computational theory and hardware capabilities.

Overall, the paper illustrates the convergence of mathematical theory with computational advancements, painted clearly by the authors' successful application of SAT technologies to provide a novel proof for a long-standing open problem.