Improved Roundtrip Spanners, Emulators, and Directed Girth Approximation

Roundtrip spanners are the analog of spanners in directed graphs, where the roundtrip metric is used as a notion of distance. Recent works have shown existential results of roundtrip spanners nearly matching the undirected case, but the time complexity for constructing roundtrip spanners is still widely open. This paper focuses on developing fast algorithms for roundtrip spanners and related problems. For any $n$-vertex directed graph $G$ with $m$ edges (with non-negative edge weights), our results are as follows: - 3-roundtrip spanner faster than APSP: We give an $\tilde{O}(m\sqrt{n})$-time algorithm that constructs a roundtrip spanner of stretch $3$ and optimal size $O(n^{3/2})$. Previous constructions of roundtrip spanners of the same size either required $\Omega(nm)$ time [Roditty, Thorup, Zwick SODA'02; Cen, Duan, Gu ICALP'20], or had worse stretch $4$ [Chechik and Lifshitz SODA'21]. - Optimal roundtrip emulator in dense graphs: For integer $k\ge 3$, we give an $O(kn^2\log n)$-time algorithm that constructs a roundtrip \emph{emulator} of stretch $(2k-1)$ and size $O(kn^{1+1/k})$, which is optimal for constant $k$ under Erd\H{o}s' girth conjecture. Previous work of [Thorup and Zwick STOC'01] implied a roundtrip emulator of the same size and stretch, but it required $\Omega(nm)$ construction time. Our improved running time is near-optimal for dense graphs. - Faster girth approximation in sparse graphs: We give an $\tilde{O}(mn^{1/3})$-time algorithm that $4$-approximates the girth of a directed graph. This can be compared with the previous $2$-approximation algorithm in $\tilde{O}(n^2, m\sqrt{n})$ time by [Chechik and Lifshitz SODA'21]. In sparse graphs, our algorithm achieves better running time at the cost of a larger approximation ratio.Comment: To appear in SODA 202

Roundtrip Spanners with (2k-1) Stretch

A roundtrip spanner of a directed graph $G$ is a subgraph of $G$ preserving roundtrip distances approximately for all pairs of vertices. Despite extensive research, there is still a small stretch gap between roundtrip spanners in directed graphs and undirected graphs. For a directed graph with real edge weights in $[1,W]$, we first propose a new deterministic algorithm that constructs a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log (nW))$ edges for every integer $k> 1$, then remove the dependence of size on $W$ to give a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log n)$ edges. While keeping the edge size small, our result improves the previous $2k+\epsilon$ stretch roundtrip spanners in directed graphs [Roditty, Thorup, Zwick'02; Zhu, Lam'18], and almost matches the undirected $(2k-1)$-spanner with $O(n^{1+1/k})$ edges [Alth\"ofer et al. '93] when $k$ is a constant, which is optimal under Erd\"os conjecture.Comment: 12 page

Conditionally Optimal Approximation Algorithms for the Girth of a Directed Graph

It is known that a better than $2$-approximation algorithm for the girth in dense directed unweighted graphs needs $n^{3-o(1)}$ time unless one uses fast matrix multiplication. Meanwhile, the best known approximation factor for a combinatorial algorithm running in $O(mn^{1-\epsilon})$ time (by Chechik et al.) is $3$. Is the true answer $2$ or $3$? The main result of this paper is a (conditionally) tight approximation algorithm for directed graphs. First, we show that under a popular hardness assumption, any algorithm, even one that exploits fast matrix multiplication, would need to take at least $mn^{1-o(1)}$ time for some sparsity $m$ if it achieves a $(2-\epsilon)$-approximation for any $\epsilon>0$. Second we give a $2$-approximation algorithm for the girth of unweighted graphs running in $\tilde{O}(mn^{3/4})$ time, and a $(2+\epsilon)$-approximation algorithm (for any $\epsilon>0$) that works in weighted graphs and runs in $\tilde{O}(m\sqrt n)$ time. Our algorithms are combinatorial. We also obtain a $(4+\epsilon)$-approximation of the girth running in $\tilde{O}(mn^{\sqrt{2}-1})$ time, improving upon the previous best $\tilde{O}(m\sqrt n)$ running time by Chechik et al. Finally, we consider the computation of roundtrip spanners. We obtain a $(5+\epsilon)$-approximate roundtrip spanner on $\tilde{O}(n^{1.5}/\epsilon^2)$ edges in $\tilde{O}(m\sqrt n/\epsilon^2)$ time. This improves upon the previous approximation factor $(8+\epsilon)$ of Chechik et al. for the same running time.Comment: To appear in ICALP 202

Reachability Preservers: New Extremal Bounds and Approximation Algorithms

We abstract and study \emph{reachability preservers}, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph $G = (V, E)$ and a set of \emph{demand pairs} $P \subseteq V \times V$, a reachability preserver is a sparse subgraph $H$ that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an $n$-node graph and demand pairs of the form $P \subseteq S \times V$ for a small node subset $S$, there is always a reachability preserver on $O(n+\sqrt{n |P| |S|})$ edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which $O(n)$ size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the $O(n^{0.6+\varepsilon})$ of Chlamatac, Dinitz, Kortsarz, and Laekhanukit (SODA'17) to $O(n^{4/7+\varepsilon})$.Comment: SODA '1

An FPT Algorithm for Minimum Additive Spanner Problem

For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. The Minimum Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive t-spanners, the Minimum Additive t-Spanner Problem is hard to handle and hence only few results are known for it. In this paper, we study the Minimum Additive t-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to the case with both a multiplicative approximation factor ? and an additive approximation parameter ?, which we call (?, ?)-spanners

On Diameter Approximation in Directed Graphs

Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds. In directed graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since d(u,v) may not be the same as d(v,u), there are multiple ways to define the problem, the two most natural being the (one-way) diameter (max_(u,v) d(u,v)) and the roundtrip diameter (max_{u,v} d(u,v)+d(v,u)). In this paper we make progress on the outstanding open question for each of them. - We design the first algorithm for diameter in sparse directed graphs to achieve n^{1.5-?} time with an approximation factor better than 2. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication. - We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a 1.5-approximation in subquadratic time would refute the All-Nodes k-Cycle hypothesis, and any (2-?)-approximation would imply a breakthrough algorithm for approximate ?_?-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH