Advanced Kernel Search approach for the MST Problem with conflicts involving affinity detection and initial solution construction
The Minimum Spanning Tree Problem with Conflicts consists in finding the minimum conflict-free spanning tree of a graph, i.e., the spanning tree of minimum cost, including no pairs of edges that are in conflict. In this paper, we solve this problem using an enhanced Kernel Search method, which iteratively solves refined problem restrictions. Our approach addresses two central open questions in the kernel search literature: (1) how to determine the affinity between variables to ensure that the restricted problem contains variables that are as compatible as possible, meaning they are more likely to appear together in a feasible solution, and (2) how to construct an initial feasible solution quickly. To this end, we integrate the computation of independent sets from the conflict graph within the algorithm to detect affinities and effectively manage conflicts. Furthermore, we heuristically construct an initial starting point, significantly accelerating the computational process. Although our methodology is designed for MSTC, its principles could be extended to other combinatorial optimization problems with conflicts. Experimental results on benchmark instances demonstrate the efficiency and competitiveness of our approach compared to existing methods in the literature, achieving 17 new best-known values.