Planar Length-Constrained Minimum Spanning Trees

TL;DR

It is shown that any algorithm on general graphs for length-constrained MST cannot achieve an approximation of O\left(\log ^{2-\epsilon} n\right) for any constant $\epsilon>0$ under standard complexity assumptions; as such, the results separate the approximability of length-constrained MST in planar and general graphs.

Abstract

In length-constrained minimum spanning tree (MST) we are given an $n$-node graph $G = (V,E)$ with edge weights $w : E \to \mathbb{Z}_{\geq 0}$ and edge lengths $l: E \to \mathbb{Z}_{\geq 0}$ along with a root node $r \in V$ and a length-constraint $h \in \mathbb{Z}_{\geq 0}$. Our goal is to output a spanning tree of minimum weight according to $w$ in which every node is at distance at most $h$ from $r$ according to $l$. We give a polynomial-time algorithm for planar graphs which, for any constant $ε> 0$, outputs an $O\left(\log^{1+ε} n\right)$-approximate solution with every node at distance at most $(1+ε)h$ from $r$ for any constant $ε> 0$. Our algorithm is based on new length-constrained versions of classic planar separators which may be of independent interest. Additionally, our algorithm works for length-constrained Steiner tree. Complementing this, we show that any algorithm on general graphs for length-constrained MST in which nodes are at most $2h$ from $r$ cannot achieve an approximation of $O\left(\log ^{2-ε} n\right)$ for any constant $ε> 0$ under standard complexity assumptions; as such, our results separate the approximability of length-constrained MST in planar and general graphs.

Figures (9)

• Figure 1: A graph in which no spanning tree contains only $h$-length shortest paths from the root $r$ for any $h, W > 0$. Edge lengths in black; edge weights in blue; all unlabeled edge lengths and weights $0$. In \ref{['sfig:noST2']} the unique $r$ to $a$$h$-length shortest path is included, preventing the inclusion of the unique $r$ to $b$$h$-length shortest path. \ref{['sfig:noST3']} gives the flipped case.
• Figure 2: A summary of work on length-constrained MST.
• Figure 3: A step of our algorithm to compute a length-constrained division. We're given a length-constrained region $(H,L_H)$ where $L_H$ contains the bolded edges/vertices (\ref{['subfig:region']}). We take a separator $P$ which contains the pink edges/vertices, $C$ is the union of $P$ and the dotted pink edge $e_C$, and $H_A$ (resp. $H_B$) contains the green (resp. blue) edges/vertices (\ref{['subfig:cyclesep-region']}). We recurse on $(H_A,P\cup(H_A\cap L_H)),(H_B,P\cup(H_B\cap L_H))$ where $P\cup(H_A\cap L_H)),P\cup(H_B\cap L_H))$ contain the bolded edges/vertices (\ref{['subfig:inside-region', 'subfig:outside-region']}).
• Figure 4: A subtree $T'_i$ rooted at $r'_i$. A $u,v$ path in $T'_i$ at worst uses $\frac{h}{4\beta}$ to go from $u$ to a child $w$ of $r'_i$, $\frac{h}{4\beta}$ to go from $w$ to $r'_i$, $\frac{h}{4\beta}$ to go from $r'_i$ to a child $x$ of $r'_i$, and $\frac{h}{4\beta}$ to go from $x$ to $v$.
• Figure 5: Graph $F\left(H,H',g,g'\right)$, where $H$ is the clear oval, where $\tilde{r}$ is the bolded blue vertex, the pieces of $\mathcal{P}$ are the red dotted ovals, the pieces of $\mathcal{P}'$ are the green dotted ovals, $U$ contains the green vertices, $X$ contains the red edges, and $X'$ contains the green edges.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Informal
• Definition 3.1: Two-Sided Balanced Separator
• Definition 3.2: Length-Constrained Separator
• Theorem 3.3: Cycle Separator; see e.g. Lemma 2 of lipton1979separator
• Definition 3.4: Mixture Metric
• Lemma 3.5
• proof
• Lemma 3.6: Length-Constrained Separator Existence and Algorithm