The Steiner Path Aggregation Problem

TL;DR

The paper tackles the Steiner Path Aggregation Problem on a directed multigraph with a specified root and terminals, where each terminal has a monochromatic path to the root. It introduces a polynomial-time algorithm that guarantees at most $2\log_{4/3} k$ color switches per terminal, and proves this bound is tight up to constants via a $\log_2 k$ lower bound. By blending ideas reminiscent of heavy path decomposition with a Matching Based Augmentation–style iterative framework and a dependency graph that is 3-colorable, the method extends tree-based decomposition techniques to general graphs. This yields a scalable, bounded-cost Steiner in-arborescence that enables simple, robust routing from terminals to the root while maintaining provable switching costs.

Abstract

In the Steiner Path Aggregation Problem, our goal is to aggregate paths in a directed network into a single arborescence without significantly disrupting the paths. In particular, we are given a directed multigraph with colored arcs, a root, and $k$ terminals, each of which has a monochromatic path to the root. Our goal is to find an arborescence in which every terminal has a path to the root, and its path does not switch colors too many times. We give an efficient algorithm that finds such a solution with at most $2\log_{4/3}k$ color switches. Up to constant factors this is the best possible universal bound, as there are graphs requiring at least $\log_2 k$ color switches.

Figures (4)

• Figure 1: The input instance is pictured on the left. Observe that each node has a monochromatic "preferred path" to the root (indicated here by the color of the node). The underlying network after removing parallel arcs is pictured in the middle, with the heavy edges shown in bold. Each heavy path consists of a path of heavy edges and the parent light edge (if any). The right shows the subgraph chosen corresponding to the heavy path decomposition. Observe the switching cost is just 1, but a more naive choice could have incurred a switching cost of up to 3.
• Figure 2: An example of the circular dependencies that may arise while greedily choosing arcs from the proposed paths of two nodes $u$ and $v$. An algorithm might select the arcs of some prefix of terminal $u$'s proposed path (blue). If the prefix ends at some other terminal $v$, then we may switch over to $v$'s proposed path (green). But now $v$'s proposed path could intersect the already chosen prefix of $u$'s proposed path, creating a cycle. This necessitates either dropping some arcs (i.e. the one highlighted in purple) or switching to a different proposed path before they intersect, both incurring some additional switching cost.
• Figure 3: An example iteration of \ref{['alg:path-agg']}. (a) shows the state of the algorithm at the beginning of the iteration. Some nodes are inactive, shown in black, while white (non-root) nodes are active. Their active path is a solid color, and is a prefix of their dashed monochromatic path. (b) shows the maximal disjoint prefixes$P'_v$ extending the active paths after \ref{['step:maxprefix']}. (c) shows the result of \ref{['step:ext3']}, extending the paths of those nodes in $S_H$ by a single arc (purple and green here). Finally (d) gives the result of the iteration. Note the nodes of $S_H$ became inactive, and also a previously inactive path was cut through by the new blue active path, so an arc was removed.
• Figure 4: An illustration of the two cases from the proof of \ref{['claim']}.

Theorems & Definitions (9)

• Theorem 1.1
• Definition 2.1: $\alpha$-Switching Arborescence
• Definition 2.2: Steiner Path Aggregation Problem
• Lemma 2.3
• proof
• Theorem 3.1
• proof : Proof of \ref{['thm:main-restated']}
• Claim 3.2
• proof