Parameterized Spanning Tree Congestion

In this paper we study the Spanning Tree Congestion problem, where we are given a graph $G=(V,E)$ and are asked to find a spanning tree $T$ of minimum maximum congestion. Here, the congestion of an edge $e\in T$ is the number of edges $uv\in E$ such that the (unique) path from $u$ to $v$ in $T$ traverses $e$. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results. We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form ``vertex-deletion distance to class $\mathcal{C}$’’, thus obtaining W[1]-hardness for parameters more restricted than treewidth, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on graphs of modular-width $4$. Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8. Complementing the problem’s W[1]-hardness for treewidth…

💡 Research Summary

The paper conducts a comprehensive study of the Spanning Tree Congestion (STC) problem from the perspective of structural parameterized complexity. Given a connected graph G, the congestion of an edge e in a spanning tree T is the number of original edges whose unique detour in T passes through e; the congestion of T is the maximum such value, and STC asks whether there exists a spanning tree with congestion at most k. While STC is known to be NP‑complete and polynomial‑time solvable for k ≤ 3, the authors investigate how various graph parameters affect its tractability.

Hardness Results.
The central negative result is that STC is W