Optimization Techniques for Solving the Minimum Red--Blue Separating Set Problem with an Application to Human--AI Text Separation

Abstract

We study the so-called Minimum Red''Blue Separating Set Problem (MRBSSP) which arises from computational geometry. In particular, we propose a compact Set-Cover ILP formulation, safe preprocessing rules, a Greedy Randomized Adaptive Search Procedure (GRASP), and two exact approaches: one is based on Logic-Based Benders Decomposition (LBBD) approach, and the second represents a novel Branch-and-Benders-Cut (BBC) approach. These two exact approaches are compared to ILP--alongside GRASP and the Variable Neighborhood Search (VNS) from literature. Experiments on 250 random graphs with up to 500 vertices reveal complementary strengths: ILP yields the best feasible solutions, LBBD provides the strongest lower bounds and smallest mean optimality gap, and BBC certifies optimality most frequently. We further introduce an NLP application distinguishing human- and AI-generated texts through vocabulary-based graph features. On 42 instances, BBC achieves the smallest mean optimality gap, certifying 16 optima, while GRASP outperforms VNS. Structural analysis of optimal vocabulary separators identifies 113 mandatory and 151 replaceable features across 272 tests, highlighting substantial variability among optimal solutions. Overall, our results demonstrate the effectiveness of complementary optimization strategies and the interpretability of optimal vocabulary-based discrimination.