Arborescences and shortest path trees when colors matter

Abstract

We are given an edge-colored (directed or undirected) graph and our objective is to find a specific type of subgraph, like a spanning tree, an arborescence, a single-source shortest path tree, a perfect matching etc., with constraints on the number of edges of each color. Some of these problems, like color-constrained spanning tree, have elegant solutions and some of them, like color-constrained perfect matching, are longstanding open questions. In this work, we study color-constrained arborescences and shortest path trees. Computing a color-constrained shortest path tree on weighted digraphs turns out to be $\mathrm{NP}$-hard in general but polynomial-time solvable when all cycles have positive weight. This polynomial-time solvability is due to the fact that the solution space is essentially the set of all color-constrained arborescences of a directed acyclic subgraph of the original graph. While finding color-constrained arborescences of digraphs is $\mathrm{NP}$-hard in general, we give an efficient algorithm when the input digraph is acyclic. Consequently, a color-constrained shortest path tree on weighted digraphs having only positive weight cycles can be efficiently computed. Our algorithm generalizes to the problem of finding a color-constrained shortest path tree with the minimum total weight. Both our algorithms use a single source shortest path algorithm and a (minimum cost) maximum flow algorithm as subroutines. By using the recent algorithm by van den Brand et al. (FOCS 2023) for these subroutines, our algorithms achieve near-linear running time when the edge weights are integral and polynomially-bounded in the size of the graph. En route, we sight nice connections to colored matroids and color-constrained bases. In fact, our approach can be adapted to find a largest common independent set of two generalized partition matroids in near-linear time.