Regression Search Problem in the Worst-Case Scenario: A NP-Hardness Proof via Reduction from Bounded (2,3)-SAT

In this article, we delve into the realm of Regression Search Problems (RSP) and their complexity under different scenarios. We explore how a problem’s difficulty can be quantified using NP-hardness and polynomial reductions, shedding light on the inherent challenges involved in solving RSP instances.
To begin with, we define RSP and its variations, including the costs of queries, wrong results, and finding bugged vertices. We then dive into the details of NP-hardness and polynomial reductions, explaining how these concepts help establish the computational hardness of RSP.
One of the key insights from our analysis is that RSP can be NP-hard even when restricted to binary DAGs, thanks to a reduction from Bounded (2,3)-SAT, a problem proven to be NP-complete in [15]. This demonstrates the intricate connections between different problems in computer science and their implications for understanding complex algorithms.
Our analysis also uncovers a new algorithm called golden bisect, which improves upon the number of queries required for the worst-case scenario compared to git bisect. Golden bisect is designed as a slight modification of git bisect, with the key difference being its ability to refrain from querying vertices with excessively low scores.
Throughout our discussion, we strive to demystify complex concepts by using everyday language and engaging metaphors or analogies. Our goal is to provide a concise and accessible summary of the article’s findings, while still capturing their essence without oversimplifying. We hope that this summary will serve as a valuable resource for those seeking to deepen their understanding of Regression Search Problems and their intricate relationship with computational complexity.