Automated Discovery of Branching Rules with Optimal Complexity for the Maximum Independent Set Problem (2412.07685v1)

Abstract: The branching algorithm is a fundamental technique for designing fast exponential-time algorithms to solve combinatorial optimization problems exactly. It divides the entire solution space into independent search branches using predetermined branching rules, and ignores the search on suboptimal branches to reduce the time complexity. The complexity of a branching algorithm is primarily determined by the branching rules it employs, which are often designed by human experts. In this paper, we show how to automate this process with a focus on the maximum independent set problem. The main contribution is an algorithm that efficiently generate optimal branching rules for a given sub-graph with tens of vertices. Its efficiency enables us to generate the branching rules on-the-fly, which is provably optimal and significantly reduces the number of branches compared to existing methods that rely on expert-designed branching rules. Numerical experiment on 3-regular graphs shows an average complexity of O(1.0441^n) can be achieved, better than any previous methods.

• The paper introduces an automated approach to discover optimal branching rules specifically tailored to graph subgraphs for the Maximum Independent Set problem.
• This method frames the branching rule discovery as a solvable Weighted Minimum Set Covering problem using integer programming techniques.
• Numerical results show the method achieves an improved average complexity of O(1.0441^n) on 3-regular graphs, outperforming existing techniques.

Automated Discovery of Branching Rules with Optimal Complexity for the Maximum Independent Set Problem

This paper addresses a significant challenge within the domain of combinatorial optimization, specifically with regards to the Maximum Independent Set (MIS) problem, which is classified as NP-complete. The focus is the enhancement of traditional branching algorithms, which are pivotal in solving such NP-hard problems. The authors present a novel approach to automatically generate optimal branching rules tailored to the unique characteristics of a given subgraph. This approach offers an improvement in computational complexity and potential applicability to a broad range of optimization problems.

The MIS problem is a foundational challenge in graph theory, where the objective is to find the largest set of vertices such that no two vertices are adjacent. Given its NP-complete nature, efficiently solving the MIS problem remains an open challenge, particularly in terms of reducing the exponential time complexity to a feasible level.

The authors propose a paradigm shift from relying on predefined human-designed branching rules to leveraging computational power for the real-time generation of optimal rules that are specific to each subgraph encountered during the problem-solving process. This is achieved by framing the branching rule problem as a Weighted Minimum Set Covering (WMSC) problem, which can be efficiently solved with integer programming techniques.

Significant numerical results are reported when applying the method to 3-regular graphs, where the algorithm demonstrates an average complexity of $O(1.0441^n)$, outperforming existing techniques that achieve $O(1.0487^n)$. This improvement is substantiated empirically across multiple graph types including Erdos-Renyi and geometric graphs, confirming the robustness and generalizability of the proposed method.

Moreover, the paper exploits the relationship between the MIS problem and theorem proving, showcasing how optimal branching rule discovery can reduce human effort in logical reasoning and proof generation. This aspect highlights a potential application of the method beyond mere optimization problems, extending to areas such as automated theorem proving where propositional logic plays a central role.

The implications of this research are manifold. Practically, the ability to generate optimal branching rules dynamically can lead to more efficient algorithms for numerous NP-hard problems, not just the MIS problem. Theoretically, the approach provides a framework for rethinking how dynamic rule generation can influence problem complexity in combinatorial optimization, potentially opening avenues for further research into automated rule optimization in other areas.

Future research can expand upon this framework by exploring modifications to the vertex selection strategy, adjustment of complexity measures, or integration with machine learning models for even more nuanced rule generation. Additionally, this could be extended to a wider class of optimization problems, such as Vertex Cover and Max-SAT, which share underlying algorithms and complexity challenges with the MIS problem.

In summary, the paper presents a methodological advancement in the automated discovery of branching rules for the MIS problem, contributing both improved performance metrics and theoretical insights into optimization algorithm design. With the full potential of automatized rule discovery yet to be explored, this work sets the stage for further innovations in the field of combinatorial optimization, theorem proving, and beyond.