Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts

We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Omega(n^{1/6}), and that any O(m)-size shortcut set cannot bring it below Omega(n^{1/11}). These improve Hesse\u27s [Hesse, 2003] lower bound of Omega(n^{1/17}). By combining these constructions with Abboud and Bodwin\u27s [Abboud and Bodwin, 2017] edge-splitting technique, we get additive stretch lower bounds of +Omega(n^{1/13}) for O(n)-size spanners and +Omega(n^{1/18}) for O(n)-size emulators. These improve Abboud and Bodwin\u27s +Omega(n^{1/22}) lower bounds

Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product

We obtain improved lower bounds for additive spanners, additive emulators, and diameter-reducing shortcut sets. Spanners and emulators are sparse graphs that approximately preserve the distances of a given graph. A shortcut set is a set of edges that when added to a directed graph, decreases its diameter. The previous best known lower bounds for these three structures are given by Huang and Pettie [SWAT 2018]. For $O(n)$-sized spanners, we improve the lower bound on the additive stretch from $\Omega(n^{1/11})$ to $\Omega(n^{2/21})$. For $O(n)$-sized emulators, we improve the lower bound on the additive stretch from $\Omega(n^{1/18})$ to $\Omega(n^{2/29})$. For $O(m)$-sized shortcut sets, we improve the lower bound on the graph diameter from $\Omega(n^{1/11})$ to $\Omega(n^{1/8})$. Our key technical contribution, which is the basis of all of our bounds, is an improvement of a graph product known as an alternation product.Comment: To appear in SODA 202

Towards Bypassing Lower Bounds for Graph Shortcuts

For a given (possibly directed) graph G, a hopset (a.k.a. shortcut set) is a (small) set of edges whose addition reduces the graph diameter while preserving desired properties from the given graph G, such as, reachability and shortest-path distances. The key objective is in optimizing the tradeoff between the achieved diameter and the size of the shortcut set (possibly also, the distance distortion). Despite the centrality of these objects and their thorough study over the years, there are still significant gaps between the known upper and lower bound results. A common property shared by almost all known shortcut lower bounds is that they hold for the seemingly simpler task of reducing the diameter of the given graph, D_G, by a constant additive term, in fact, even by just one! We denote such restricted structures by (D_G-1)-diameter hopsets. In this paper we show that this relaxation can be leveraged to narrow the current gaps, and in certain cases to also bypass the known lower bound results, when restricting to sparse graphs (with O(n) edges): - {Hopsets for Directed Weighted Sparse Graphs.} For every n-vertex directed and weighted sparse graph G with D_G ? n^{1/4}, one can compute an exact (D_G-1)-diameter hopset of linear size. Combining this with known lower bound results for dense graphs, we get a separation between dense and sparse graphs, hence shortcutting sparse graphs is provably easier. For reachability hopsets, we can provide (D_G-1)-diameter hopsets of linear size, for sparse DAGs, already for D_G ? n^{1/5}. This should be compared with the diameter bound of O?(n^{1/3}) [Kogan and Parter, SODA 2022], and the lower bound of D_G = n^{1/6} by [Huang and Pettie, {SIAM} J. Discret. Math. 2018]. - {Additive Hopsets for Undirected and Unweighted Graphs.} We show a construction of +24 additive (D_G-1)-diameter hopsets with linear number of edges for D_G ? n^{1/12} for sparse graphs. This bypasses the current lower bound of D_G = n^{1/6} obtained for exact (D_G-1)-diameter hopset by [HP\u2718]. For general graphs, the bound becomes D_G ? n^{1/6} which matches the lower bound of exact (D_G-1) hopsets implied by [HP\u2718]. We also provide new additive D-diameter hopsets with linear size, for any given diameter D. Altogether, we show that the current lower bounds can be bypassed by restricting to sparse graphs (with O(n) edges). Moreover, the gaps are narrowed significantly for any graph by allowing for a constant additive stretch

Simpler and Higher Lower Bounds for Shortcut Sets

We provide a variety of lower bounds for the well-known shortcut set problem: how much can one decrease the diameter of a directed graph on $n$ vertices and $m$ edges by adding $O(n)$ or $O(m)$ of shortcuts from the transitive closure of the graph. Our results are based on a vast simplification of the recent construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an $\widetilde{\Omega}(n^{1/4})$ lower bound for the $O(n)$-sized shortcut set problem. We highlight that our simplification completely removes the use of the convex sets by B\'ar\'any and Larman [Math. Ann. 1998] used in all previous lower bound constructions. Our simplification also removes the need for randomness and further removes some log factors. This allows us to generalize the construction to higher dimensions, which in turn can be used to show the following results. For $O(m)$-sized shortcut sets, we show an $\Omega(n^{1/5})$ lower bound, improving on the previous best $\Omega(n^{1/8})$ lower bound. For all $\varepsilon > 0$, we show that there exists a $\delta > 0$ such that there are $n$-vertex $O(n)$-edge graphs $G$ where adding any shortcut set of size $O(n^{2-\varepsilon})$ keeps the diameter of $G$ at $\Omega(n^\delta)$. This improves the sparsity of the constructed graph compared to a known similar result by Hesse [SODA 2003]. We also consider the sourcewise setting for shortcut sets: given a graph $G=(V,E)$, a set $S\subseteq V$, how much can we decrease the sourcewise diameter of $G$, $\max_{(s, v) \in S \times V, \text{dist}(s, v) < \infty} \text{dist}(s,v)$ by adding a set of edges $H$ from the transitive closure of $G$? We show that for any integer $d \ge 2$, there exists a graph $G=(V, E)$ on $n$ vertices and $S \subseteq V$ with $|S| = \widetilde{\Theta}(n^{3/(d+3)})$, such that when adding $O(n)$ or $O(m)$ shortcuts, the sourcewise diameter is $\widetilde{\Omega}(|S|^{1/3})$.Comment: To appear in SODA 2024. Abstract shortened to fit arXiv requirement

Closing the Gap Between Directed Hopsets and Shortcut Sets

For an n-vertex directed graph $G = (V,E)$, a $\beta$-\emph{shortcut set} $H$ is a set of additional edges $H \subseteq V \times V$ such that $G \cup H$ has the same transitive closure as $G$, and for every pair $u,v \in V$, there is a $uv$-path in $G \cup H$ with at most $\beta$ edges. A natural generalization of shortcut sets to distances is a $(\beta,\epsilon)$-\emph{hopset} $H \subseteq V \times V$, where the requirement is that $H$ and $G \cup H$ have the same shortest-path distances, and for every $u,v \in V$, there is a $(1+\epsilon)$-approximate shortest path in $G \cup H$ with at most $\beta$ edges. There is a large literature on the tradeoff between the size of a shortcut set / hopset and the value of $\beta$. We highlight the most natural point on this tradeoff: what is the minimum value of $\beta$, such that for any graph $G$, there exists a $\beta$-shortcut set (or a $(\beta,\epsilon)$-hopset) with $O(n)$ edges? Not only is this a natural structural question in its own right, but shortcuts sets / hopsets form the core of many distributed, parallel, and dynamic algorithms for reachability / shortest paths. Until very recently the best known upper bound was a folklore construction showing $\beta = O(n^{1/2})$, but in a breakthrough result Kogan and Parter [SODA 2022] improve this to $\beta = \tilde{O}(n^{1/3})$ for shortcut sets and $\tilde{O}(n^{2/5})$ for hopsets. Our result is to close the gap between shortcut sets and hopsets. That is, we show that for any graph $G$ and any fixed $\epsilon$ there is a $(\tilde{O}(n^{1/3}),\epsilon)$ hopset with $O(n)$ edges. More generally, we achieve a smooth tradeoff between hopset size and $\beta$ which exactly matches the tradeoff of Kogan and Parter for shortcut sets (up to polylog factors). Using a very recent black-box reduction of Kogan and Parter, our new hopset implies improved bounds for approximate distance preservers.Comment: Abstract shortened to meet arXiv requirements, v2: fixed a typ

Near-Optimal Distance Emulator for Planar Graphs

Given a graph G and a set of terminals T, a distance emulator of G is another graph H (not necessarily a subgraph of G) containing T, such that all the pairwise distances in G between vertices of T are preserved in H. An important open question is to find the smallest possible distance emulator. We prove that, given any subset of k terminals in an n-vertex undirected unweighted planar graph, we can construct in O~(n) time a distance emulator of size O~(min(k^2,sqrt{k * n})). This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among k terminals in O~(n) time when k=O(n^{1/3})

Bridge Girth: A Unifying Notion in Network Design

A classic 1993 paper by Alth\H{o}fer et al. proved a tight reduction from spanners, emulators, and distance oracles to the extremal function $\gamma$ of high-girth graphs. This paper initiated a large body of work in network design, in which problems are attacked by reduction to $\gamma$ or the analogous extremal function for other girth concepts. In this paper, we introduce and study a new girth concept that we call the bridge girth of path systems, and we show that it can be used to significantly expand and improve this web of connections between girth problems and network design. We prove two kinds of results: 1) We write the maximum possible size of an $n$-node, $p$-path system with bridge girth $>k$ as $\beta(n, p, k)$, and we write a certain variant for "ordered" path systems as $\beta^*(n, p, k)$. We identify several arguments in the literature that implicitly show upper or lower bounds on $\beta, \beta^*$, and we provide some polynomially improvements to these bounds. In particular, we construct a tight lower bound for $\beta(n, p, 2)$, and we polynomially improve the upper bounds for $\beta(n, p, 4)$ and $\beta^*(n, p, \infty)$. 2) We show that many state-of-the-art results in network design can be recovered or improved via black-box reductions to $\beta$ or $\beta^*$. Examples include bounds for distance/reachability preservers, exact hopsets, shortcut sets, the flow-cut gaps for directed multicut and sparsest cut, an integrality gap for directed Steiner forest. We believe that the concept of bridge girth can lead to a stronger and more organized map of the research area. Towards this, we leave many open problems, related to both bridge girth reductions and extremal bounds on the size of path systems with high bridge girth

Folklore Sampling is Optimal for Exact Hopsets: Confirming the n\sqrt{n} Barrier

For a graph $G$, a $D$-diameter-reducing exact hopset is a small set of additional edges $H$ that, when added to $G$, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most $D$ edges. A shortcut set is the analogous concept for reachability. These objects have been studied since the early '90s due to applications in parallel, distributed, dynamic, and streaming graph algorithms. For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA '22] and Bernstein and Wein [SODA '23] have finally improved over the folklore diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+\epsilon)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued? We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetilde{\Omega}(n^{1/2})$. This improves on the previous lower bound of $\widetilde{\Omega}(n^{1/3})$ by Kogan and Parter [FOCS '22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetilde{\Omega}(n^{1/4})$. This improves on the previous lower bound of $\Omega(n^{1/6})$ by Huang and Pettie [SIAM J. Disc. Math. '18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of $p$; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of $p \in [1, n^2]$