Two Complexity Results on Spanning-Tree Congestion Problems

TL;DR

It is proved that the decision version of the spanning-tree congestion problem, given an integer $K$, is polynomial-time solvable for $K$-edge-connected graphs.

Abstract

In the spanning-tree congestion problem ($\mathsf{STC}$), we are given a graph $G$, and the objective is to compute a spanning tree of $G$ that minimizes the maximum edge congestion. While $\mathsf{STC}$ is known to be $\mathbb{NP}$-hard, even for some restricted graph classes, several key questions regarding its computational complexity remain open, and we address some of these in our paper. (i) For graphs of degree at most $d$, it is known that $\mathsf{STC}$ is $\mathbb{NP}$-hard when $d\ge 8$. We provide a complete resolution of this variant, by showing that $\mathsf{STC}$ remains $\mathbb{NP}$-hard for each degree bound $d\ge 3$. (ii) In the decision version of $\mathsf{STC}$, given an integer $K$, the goal is to determine whether the congestion of $G$ is at most $K$. We prove that this variant is polynomial-time solvable for $K$-edge-connected graphs.

Figures (11)

• Figure 1: A $4$-edge-connected multigraph and all its $4$-cuts. Parallel lines represent double parallel edges. Each vertex $v_i$ is identified by its index $i$. For $X = { \left\{ { v_{4},v_{5},v_{6},v_{7},v_{8} } \right\} }$ and $Y = { \left\{ {v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}} \right\} }$, cuts ${\partial X}$, ${\partial Y}$ cross.
• Figure 2: A cactus representation of the graph from Figure \ref{['fig: 4-cuts_example_graph']}. The numbers represent indices of vertices in $G$ that are mapped by $\phi$ to the corresponding node in ${\mathfrak{C}}_G$. Note that the pre-image of a node in ${\mathfrak{C}}_G$ could be empty.
• Figure 3: On the left, an illustration of Lemma \ref{['lem: K-cuts, type 1 node']}. On the right, an illustration of Lemma \ref{['lem: K-cuts, type 0 node']}. In both examples $K=8$ and $d=4$. Edges in $\widetilde{T}$ that cross the basic $K$-cuts ${\partial Z_i}$ are green (dark) and thick, non-tree edges are brown (light) and thin.
• Figure 4: An illustration of Lemma \ref{['lem: K-cuts, cycle']}, for $K=8$, $\ell = 6$ and $g=4$.
• Figure 5: An illustration of Lemma \ref{['lem:hub-cycle']}(iii), for $K=8$, $\ell = 7$ and $g=4$. Thick blue edges show the front spine $w_0,w_1,w_2$ and back spine $w_5,w_6,w_7$, with $w_7 = w_0$. For $i\in{ \left\{ {1,2,5,6} \right\} }$, the witness trees $T_i$ for $w_i$ and $Z_i$ are depicted with thin purple lines. The witness trees for $w_2$ and $Z_3$, and for $w_5$ and $Z_4$ are depicted with thin blue lines. Thin brown lines are non-tree edges in the cuts ${\partial Z_i}$.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 4.1
• Theorem 4.2
• proof
• Lemma 4.6
• proof
• Lemma 4.7
• proof
• Corollary 4.8