Arborescences and Shortest Path Trees when Colors Matter
P. S. Ardra, Jasine Babu, Kritika Kashyap, R. Krithika, Sreejith K. Pallathumadam, Deepak Rajendraprasad
TL;DR
This paper investigates color-constrained subgraph problems, focusing on arborescences and shortest path trees in $q$-colored graphs. It shows CC-SPT is $NP$-hard in general but becomes tractable when all cycles have positive weight by reducing CC-SPT to CC-ARB on the $s$-shortest-path graph $G_s$, and extends this to minimum-weight variants. A flow-based framework, CC-ARB-Flow, computes $oldsymbol{}$-colored arborescences on DAGs, with efficient special cases including a linear-time solution for red-blue graphs and reductions to $2$-Matroid Intersection for DAGs. The paper also analyzes CC-SP and VCC-SP, establishing their equivalence under polynomial-time reductions and detailing hardness results via graph transformations. Overall, the work bridges color-constrained subgraph problems with matroid theory and flows, delivering near-linear-time algorithms under positive-cycle assumptions and outlining rich avenues for parameterized complexity research.
Abstract
Color-constrained subgraph problems are those where we are given an edge-colored (directed or undirected) graph and the task 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 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 arborescence of digraphs is NP-hard in general, we give efficient algorithms when the input graph is acyclic. Consequently, a color-constrained shortest path tree on weighted digraphs having only positive weight cycles can be efficiently computed. Our algorithms also generalize to the problem of finding a color-constrained shortest path tree with minimum total weight. En route, we sight nice connections to colored matroids and color-constrained bases.