A Polylogarithmic Approximation for Directed Steiner Forest in Planar Digraphs

TL;DR

This work shows that in planar digraphs, an important and useful class of graphs in both theory and practice, DSF is much more tractable, and obtains an O(\log^6 k)-approximation algorithm via the junction tree technique.

Abstract

We consider Directed Steiner Forest (DSF), a fundamental problem in network design. The input to DSF is a directed edge-weighted graph $G = (V, E)$ and a collection of vertex pairs $\{(s_i, t_i)\}_{i \in [k]}$. The goal is to find a minimum cost subgraph $H$ of $G$ such that $H$ contains an $s_i$-$t_i$ path for each $i \in [k]$. DSF is NP-Hard and is known to be hard to approximate to a factor of $Ω(2^{\log^{1 - ε}(n)})$ for any fixed $ε> 0$ [DK'99]. DSF admits approximation ratios of $O(k^{1/2 + ε})$ [CEGS'11] and $O(n^{2/3 + ε})$ [BBMRY'13]. In this work we show that in planar digraphs, an important and useful class of graphs in both theory and practice, DSF is much more tractable. We obtain an $O(\log^6 k)$-approximation algorithm via the junction tree technique. Our main technical contribution is to prove the existence of a low density junction tree in planar digraphs. To find an approximate junction tree we rely on recent results on rooted directed network design problems [FM'23, CJKZZ'24], in particular, on an LP-based algorithm for the Directed Steiner Tree problem [CJKZZ'24]. Our work and several other recent ones on algorithms for planar digraphs [FM'23, KS'21, CJKZZ'24] are built upon structural insights on planar graph reachability and shortest path separators [Thorup'04].

Figures (4)

• Figure 1: Example of a two-layered digraph. Bolded edges form the two-layered spanning tree; remaining edges in the graph are dashed. The two dipaths for each root to leaf path are denoted by blue and red edges: the first dipath away from the root given in blue and the second towards the root in red.
• Figure 2: Layers constructed from $G^*$. Dotted lines represent edges inside each layer, while solid lines represent edges between layers. In this example, $j$ is odd.
• Figure 3: Example of separator and resulting weakly connected components. Solid black lines denote edges in the separator $S_0$, while dashed blue lines represent edges between components and the separator. Terminals are labeled and denoted with boxes. In this example, $D_0 = \{(s_1, t_1), (s_2, t_2)\}$ since there exists an $s_1$-$t_1$ and an $s_2$-$t_2$ path through the separator. Notice that $(s_2, t_2) \in D_0$ even though $s_2$ and $t_2$ remain in the same component $C_1$.
• Figure 4: The path $P$ is given with solid black lines. Blue dashed lines represent the paths between terminals and $P$. Note that terminals can have multiple paths to/from $P$, as shown by $s_4$/$t_4$. In this example, terminal pairs $1,2,3$ all have mutually overlapping intervals and thus form a junction tree rooted at the vertex $b_1 = a_3$.

Theorems & Definitions (46)

• theorem 1
• remark 1: Node Weights
• theorem 2
• remark 2
• theorem 3
• definition 1: Junction Tree
• theorem 3
• definition 2
• remark 3
• lemma 3