Improved Online Reachability Preservers

TL;DR

The techniques give a small polynomial improvement in the current upper bounds for offline reachability preservers, and they extend to a stronger model in which they must commit to a path for all possible reachable pairs in $G$ before any demand pairs have been received.

Abstract

A reachability preserver is a basic kind of graph sparsifier, which preserves the reachability relation of an $n$-node directed input graph $G$ among a set of given demand pairs $P$ of size $|P|=p$. We give constructions of sparse reachability preservers in the online setting, where $G$ is given on input, the demand pairs $(s, t) \in P$ arrive one at a time, and we must irrevocably add edges to a preserver $H$ to ensure reachability for the pair $(s, t)$ before we can see the next demand pair. Our main results are: -- There is a construction that guarantees a maximum preserver size of $$|E(H)| \le O\left( n^{0.72}p^{0.56} + n^{0.6}p^{0.7} + n\right).$$ This improves polynomially on the previous online upper bound of $O( \min\{np^{0.5}, n^{0.5}p\}) + n$, implicit in the work of Coppersmith and Elkin [SODA '05]. -- Given a promise that the demand pairs will satisfy $P \subseteq S \times V$ for some vertex set $S$ of size $|S|=:σ$, there is a construction that guarantees a maximum preserver size of $$|E(H)| \le O\left( (npσ)^{1/2} + n\right).$$ A slightly different construction gives the same result for the setting $P \subseteq V \times S$. This improves polynomially on the previous online upper bound of $O( σn)$ (folklore). All of these constructions are polynomial time, deterministic, and they do not require knowledge of the values of $p, σ$, or $S$. Our techniques also give a small polynomial improvement in the current upper bounds for offline reachability preservers, and they extend to a stronger model in which we must commit to a path for all possible reachable pairs in $G$ before any demand pairs have been received. As an application, we improve the competitive ratio for Online Unweighted Directed Steiner Forest to $O(n^{3/5 + \varepsilon})$.

Figures (5)

• Figure 1: A directed $3$-cycle (left) and a $3$-bridge (right).
• Figure 2: The proof of Lemma \ref{['lem:pathintbound']} works by arguing that no path $\pi$ may intersect too many paths that come later in the ordering, or else two of those paths $q_1, q_2$ will share an endpoint in $S$ and thus form a forbidden $3$-bridge.
• Figure 3: There cannot be two paths $\pi_3, \pi_4$ that both intersect paths $\pi_1, \pi_2$, but where the points of intersection switch places as in this picture, or else they form a $4$-bridge (here $\pi_3$ is the river).
• Figure 4: The random subsystem $S'$. Figure based on BHT23, Figure 7.
• Figure 5: When we generate $S'$ in the online/ordered setting, it is possible for the paths intersecting $\pi_b$ to intersect each other: the river could come before the $2^{nd}$ arc, which would not violate the conditions of Lemma \ref{['lem:zprops']}.

Theorems & Definitions (71)

• Definition 1: Reachability Preservers
• Theorem 1: Online Pairwise Reachability Preservers
• Theorem 2: Online Source-Restricted Reachability Preservers
• Theorem 3: BHT23
• Theorem 4: Offline Reachability Preservers
• Theorem 5: Non-Adaptive Reachability Preservers
• Theorem 6: Offline UDSN CEGS11AB18
• Theorem 7: Online UDSN, parametrized on $p$ ECKP15
• Theorem 8: Online UDSN, parametrized on $n$
• Theorem 9: DAG Reduction (folklore)