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.

翻译：我们研究了在匹配约束下使用最少边数连接多部图各部分的问题。我们引入了互连树的概念，定义为在商图上的投影形成生成树的匹配。受化学信息学应用的启发，我们探讨了该问题的判定、计数和枚举变体。我们证明判定问题是$NP$完全的。尽管如此，它在若干结构化设定下变得可处理：关于部分数量具有固定参数可处理性，并在完全、拟完全和$t$-拟完全多部图上承认多项式或线性时间算法。我们还研究了枚举问题，为此我们设计了基于高效闪光灯搜索的算法，对完全多部图具有最优延迟，以及一种权重引导启发式方法，优先考虑低权重解并在实践中表现良好。