Complexity of Finding and Enumerating Interconnection Trees

TL;DR

This paper investigates the complexity of connecting multipartite graph parts with minimal edges under matching constraints, introducing interconnection trees and analyzing decision, counting, and enumeration problems.

Contribution

It defines interconnection trees in multipartite graphs, proves NP-completeness of the decision problem, and provides efficient algorithms for special graph classes and enumeration.

Findings

Decision problem is NP-complete.

Tractable in fixed-parameter and certain structured graphs.

Efficient enumeration algorithms with optimal delay.

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.