Minimum-weight maximal flames in general directed graphs

Determine the computational complexity of finding a minimum-weight maximal flame in a general directed graph, equivalently, a minimum-weight spanning subgraph that preserves every rooted local edge-connectivity value from the designated root.

Background

A flame is a rooted directed graph in which the in-degree of every non-root vertex equals its local edge-connectivity from the root. Lovász's theorem guarantees that every rooted directed graph has a spanning maximal flame preserving all rooted local edge-connectivity values, and a polynomial-time algorithm can find an unweighted maximal flame.

The paper studies the weighted optimization problem of finding a minimum-weight maximal flame. It establishes a strongly polynomial-time algorithm when the input digraph is acyclic, but leaves the complexity for general directed graphs unresolved. Resolving this problem would extend the known tractable cases beyond acyclic digraphs and the special case in which all rooted local edge-connectivity values are equal.