🤖 AI Summary
This study addresses the decision, counting, and enumeration problems for interconnected structures in multipartite graphs that satisfy matching constraints and whose projection forms a spanning tree—referred to as “interconnection trees.” The work formally introduces the notion of interconnection trees and proves that their decision problem is NP-complete. However, when the number of partitions is fixed, it presents a fixed-parameter tractable (FPT) algorithm and achieves polynomial- or even linear-time solutions on complete and quasi-complete multipartite graphs. Leveraging parameterized complexity analysis, graph-theoretic algorithm design, and the flashlight search framework, the paper proposes an enumeration algorithm with optimal delay and enhances its practicality through a weight-guided heuristic strategy.
📝 Abstract
We study the problem of connecting the parts of a multipartite graph using a minimum number of edges under a matching constraint. We introduce interconnection trees, defined as matchings whose projections onto the quotient graph form a spanning tree. Motivated by applications in chemoinformatics, we investigate the decision, counting, and enumeration variants of this problem.
We show that the decision problem is $NP$-complete. Nevertheless, it becomes tractable in several structured settings: it is fixed-parameter tractable in the number of parts, and admits polynomial or linear-time algorithms on complete, quasi-complete, and $t$-quasi-complete multipartite graphs. We also study enumeration, for which we design efficient flashlight-search based algorithms with optimal delay for complete multipartite graphs, and a weight-guided heuristic that prioritizes low-weight solutions and performs well in practice.