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Parameterized Spanning Tree Congestion

Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, Daniel Vaz

TL;DR

The paper investigates Spanning Tree Congestion (STC), the problem of selecting a spanning tree that minimizes the maximum edge congestion. It delivers a comprehensive parameterized complexity landscape: STC is not FPT by treewidth via a general vertex-deletion-to-class reduction, with NP-hardness on modular-width $4$ and maximum degree $8$, while there exist explicit FPT algorithms and FPT-approximation schemes for bounded treewidth and related deletion-distance parameters. It also shows FPT results for several graph parameters, including distance to clique, vertex integrity, and feedback edge number, often via ILP reductions or kernelization. Together, these results map the tractability frontier for STC under structural graph parameters and provide practical FPT approaches and kernels for several important cases. The work connects MSO/Courcelle-type perspectives with explicit dynamic-programming and ILP-based strategies to yield concrete running times and approximation guarantees.

Abstract

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...

Parameterized Spanning Tree Congestion

TL;DR

The paper investigates Spanning Tree Congestion (STC), the problem of selecting a spanning tree that minimizes the maximum edge congestion. It delivers a comprehensive parameterized complexity landscape: STC is not FPT by treewidth via a general vertex-deletion-to-class reduction, with NP-hardness on modular-width and maximum degree , while there exist explicit FPT algorithms and FPT-approximation schemes for bounded treewidth and related deletion-distance parameters. It also shows FPT results for several graph parameters, including distance to clique, vertex integrity, and feedback edge number, often via ILP reductions or kernelization. Together, these results map the tractability frontier for STC under structural graph parameters and provide practical FPT approaches and kernels for several important cases. The work connects MSO/Courcelle-type perspectives with explicit dynamic-programming and ILP-based strategies to yield concrete running times and approximation guarantees.

Abstract

In this paper we study the Spanning Tree Congestion problem, where we are given a graph and are asked to find a spanning tree of minimum maximum congestion. Here, the congestion of an edge is the number of edges such that the (unique) path from to in traverses . 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 '', 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 . 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...

Paper Structure

This paper contains 13 sections, 9 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardness results
  4. Distance to disjoint union
  5. Modular-width
  6. Maximum degree
  7. Algorithms for Bounded Treewidth
  8. FPT Algorithm Parameterized by Treewidth and Congestion
  9. FPT Approximation and Clique-width
  10. FPT Algorithms
  11. Distance to Clique
  12. Vertex integrity
  13. Feedback Edge Number

Key Result

Theorem 1

Spanning Tree Congestion is W[1]-hard parameterized by vertex-deletion distance to disjoint union of graphs in $\mathcal{C}$, where $\mathcal{C}$ is any graph class such that, for all $p \in \mathbb{Z}^{+}$, $\mathcal{C}$ contains a connected $p$-vertex graph which can be generated in time $p^{{\cal

Figures (3)

  • Figure 1: Our results and hierarchy of the related graph parameters (see \ref{['paragraph:graph_parameters']} for their definitions). For any graph, if the parameter at the tail of an arrow is a constant, that is also the case for the one at its head. Green indicates that the problem is FPT (\ref{['thm:distance_to_clique', 'thm:vi', 'thm:fes']}), orange W[1]-hardness (\ref{['thm:disjoint-union']}), and red para-NP-hardness (\ref{['thm:hardness:modularwidth']}). We additionally show fixed-parameter tractability by $\mathrm{cw}+k$ (\ref{['thm:cw']}), para-NP-hardness by maximum degree (\ref{['thm:maxdeg']}), as well as develop an FPT-AS for $\mathrm{tw}$ (\ref{['thm:tw-apx']}). Prior to this work, it was only known that the problem is FPT by $\mathrm{tw}+k$algorithmica/BodlaenderFGOL12 and para-NP-hard by clique-width jgaa/OkamotoOUU11.
  • Figure 3: (Left) The gadget for a double-weighted edge $e = \{u,v\}$ with $(\mathrm{w}_{1}(e), \mathrm{w}_{2}(e)) = (3,5)$. There are $\mathrm{w}_{1}(e)$ ($= 3$) grids connected to $\{u,v\}$ and each grid is of size $(\mathrm{w}_{2}(e) - \mathrm{w}_{1}(e) + 1) \times (\mathrm{w}_{2}(e) - \mathrm{w}_{1}(e) + 1)$ ($= 3 \times 3$). (Center & Right) The intersection of the gadget and the spanning tree $T'$ of $G'$ obtained from a spanning tree $T$ of $G$ for the cases $e \in E(T)$ and $e \notin E(T)$: in the former case, the edges $\{v_1,v\},\{v_2,v\},\{v_3,v\}$ and some edges in $\Gamma_1$ contribute to the congestion of the $u$--$v$ path in $T'$; in the latter, only the edges $\{v_1,v\},\{v_2,v\},\{v_3,v\}$ do.
  • Figure 4: The construction of $G$, where $c_{1} = \{x_{1}, \bar{x}_{2}, x_{3}\}$, $c_{2} = \{x_{1}, x_{2}, \bar{x}_{3}\}$, $c_{3} = \{\bar{x}_{1}, \bar{x}_{2}, x_{3}\}$, and $c_{4} = \{\bar{x}_{1}, x_{2}, \bar{x}_{3}\}$.

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Proposition 4: algorithmica/BodlaenderFGOL12
  • Lemma 5
  • Lemma 6
  • Theorem 14
  • Theorem 20
  • Definition 21
  • Definition 22
  • ...and 1 more