Covering Approximate Shortest Paths with DAGs

TL;DR

The main upper bound is that there is a near-linear time algorithm to construct a DAG cover with Õ(m) additional edges, polylogarithmic distortion, and only O(logn) DAGs, and a lower bound showing that achieving a DAG cover with no distortion and Õ(m) additional edges requires a polynomial number of DAGs.

Abstract

We define and study analogs of probabilistic tree embedding and tree cover for directed graphs. We define the notion of a DAG cover of a general directed graph $G$: a small collection $D_1,\dots D_g$ of DAGs so that for all pairs of vertices $s,t$, some DAG $D_i$ provides low distortion for $dist(s,t)$; i.e. $ dist_G(s, t) \le \min_{i \in [g]} dist_{D_i}(s, t) \leq α\cdot dist_G(s, t)$, where $α$ is the distortion. As a trivial upper bound, there is a DAG cover with $n$ DAGs and $α=1$ by taking the shortest-paths tree from each vertex. When each DAG is restricted to be a subgraph of $G$, there is a matching lower bound (via a directed cycle) that $n$ DAGs are necessary, even to preserve reachability. Thus, we allow the DAGs to include a limited number of additional edges not in the original graph. When $n^2$ additional edges are allowed, there is a simple upper bound of two DAGs and $α=1$. Our first result is an almost-matching lower bound that even for $n^{2-o(1)}$ additional edges, at least $n^{1-o(1)}$ DAGs are needed, even to preserve reachability. However, the story is different when the number of additional edges is $\tilde{O}(m)$, a natural setting where the sparsity of the DAG collection nearly matches the original graph. Our main upper bound is that there is a near-linear time algorithm to construct a DAG cover with $\tilde{O}(m)$ additional edges, polylogarithmic distortion, and only $O(\log n)$ DAGs. This is similar to known results for undirected graphs: the well-known FRT probabilistic tree embedding implies a tree cover where both the number of trees and the distortion are logarithmic. Our algorithm also extends to a certain probabilistic embedding guarantee. Lastly, we complement our upper bound with a lower bound showing that achieving a DAG cover with no distortion and $\tilde{O}(m)$ additional edges requires a polynomial number of DAGs.

Figures (3)

• Figure 1: On the left is a directed graph $G$, with strongly connected components shown in blue. On the right is one of the two DAGs constructed in our reachability-preserving DAG cover of $G$ in Observation \ref{['obs:m_reach']}. The other DAG in our cover can be obtained by reversing all the (green) edges between vertices in the same SCC of $G$, and removing all of the (red) edges between vertices in different SCCs of $G$.
• Figure 2: A visualization of the different cases in \ref{['clm:small_paths']}. These cases correspond to three possibilities of how edges entering and exiting a clique $K_2^v$ in $G^*$ are attached to vertices in $K_2^v$.
• Figure 3: A visualization of path $\pi_{s, t}$ in subgraph $G_i$ of $G$. We upper bound the distance from $s$ to $t$ in DAG $D_1$ by upper bounding the distance from $x_i$ to $y_i$ in DAG $D_1$ for all $i \in [1, q-2]$.

Theorems & Definitions (113)

• Definition 1.1: DAG cover
• Theorem 1.2
• Theorem 1.3
• proof : Proof sketch of Observation \ref{['obs:m_reach']}
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 4.1
• Lemma 4.1: cf. Theorem 13 of MR4716717