arXiv subjects

Greg Bodwin

Publications and source records attributed to Greg Bodwin.

At least 19 recordsLinked to original sources

arXiv2026

Unconditional Lower Bounds for Degree Fault Tolerant Spanners

We study multiplicative graph spanners in the $f$-degree fault tolerant ($f$-DFT) model, in which the spanner must approximately preserve distances even after any subset of edges of maximum degree $f$ temporarily "fails" and is removed from the graph. We prove that there are $n$-node lower bound graphs for which any $f$-DFT $(2k-1)$-stretch spanner $H$ must have size $$|E(H)| \ge \Omega\left( f^{1-1/k} n^{1+1/k}\right).$$ This matches a lower bound that was previously only known to hold conditionally, under the 1963 girth conjecture of Erd\H{o}s. It also matches the current upper bounds, up to a factor of $\texttt{exp}(k)$. Our proof is an analysis of the so-called Wenger graphs (J. Comb. Theory 1991), via their recent reinterpretation by Szab\'o and by Conlon (Am. Math. Monthly 2021).

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arXiv2026

Improved Upper Bounds for the Directed Flow-Cut Gap

We prove that the flow-cut gap for $n$-node directed graphs is at most $n^{1/3 + o(1)}$. This is the first improvement since a previous upper bound of $\widetilde{O}(n^{11/23})$ by Agarwal, Alon, and Charikar (STOC '07), and it narrows the gap to the current lower bound of $\widetilde{\Omega}(n^{1/7})$ by Chuzhoy and Khanna (JACM '09). We also show an upper bound on the directed flow-cut gap of $W^{1/2}n^{o(1)}$, where $W$ is the sum of the minimum fractional cut weights. As an auxiliary contribution, we significantly expand the network of reductions among various versions of the directed flow-cut gap problem. In particular, we prove near-equivalence between the edge and vertex directed flow-cut gaps, and we show that when parametrizing by $W$, one can assume unit capacities and uniform fractional cut weights without loss of generality.

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arXiv2025

Greedy Algorithms for Shortcut Sets and Hopsets

For many popular graph metric sparsifiers, such as spanners, emulators, and preservers, simple and elegant greedy algorithms are known that achieve state-of-the-art or existentially optimal tradeoffs between size and quality. The goal of this paper is to develop and analyze comparable greedy algorithms for nearby objects in graph metric augmentation. We show the following: - A simple greedy algorithm for shortcut sets achieves the state-of-the-art size/hopbound tradeoff recently proved by Kogan and Parter (2022), up to $O(\log n)$ factors in the size. Moreover, with an additional preprocessing step, the greedy algorithm subpolynomially improves on the previous size bounds in some range of parameters. - The same greedy algorithm was already known to be existentially optimal for the size/hopbound tradeoff for hopsets, by an analysis of Berman, Raskhodnikova, and Ruan (2010) introduced for transitive-closure spanners. We provide a completely different analysis showing that the algorithm is also existentially optimal (up to $O(\log n)$ factors) for the matching hopset problem, in which one has a budget of roughly $O(m)$ additional edges (for an $m$-edge input graph).

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arXiv2025

Simple Length-Constrained Expander Decompositions

Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an $(h, s)$-length $\phi$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ while using each edge to an extent at most $1/\phi$. Prior work showed that every $n$-node and $m$-edge graph admits an $(h, s)$-length $\phi$-expander decomposition of size $\log n \cdot s n^{O(1/s)} \cdot \phi m$. In this work, we give a simple proof of the existence of $(h, s)$-length $\phi$-expander decompositions with an improved size of $s n^{O(1/s)}\cdot \phi m$. Our proof is a straightforward application of the fact that the union of sparse length-constrained cuts is itself a sparse length-constrained cut. In deriving our result, we improve the loss in sparsity when taking the union of sparse length-constrained cuts from $\log ^3 n\cdot s^3 n^{O(1/s)}$ to $s\cdot n^{O(1/s)}$.

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arXiv2025

Notes on the Linear Algebraic View of Regularity Lemmas

When regularity lemmas were first developed in the 1970s, they were described as results that promise a partition of any graph into a ``small'' number of parts, such that the graph looks ``similar'' to a random graph on its edge subsets going between parts. Regularity lemmas have been repeatedly refined and reinterpreted in the years since, and the modern perspective is that they can instead be seen as purely linear-algebraic results about sketching a large, complicated matrix with a smaller, simpler one. These matrix sketches then have a nice interpretation about partitions when applied to the adjacency matrix of a graph. In these notes we will develop regularity lemmas from scratch, under the linear-algebraic perspective, and then use the linear-algebraic versions to derive the familiar graph versions. We do not assume any prior knowledge of regularity lemmas, and we recap the relevant linear-algebraic definitions as we go, but some comfort with linear algebra will definitely be helpful to read these notes.

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arXiv2025

Multiplicative Spanners in Minor-Free Graphs

In FOCS 2017, Borradaille, Le, and Wulff-Nilsen addressed a long-standing open problem by proving that minor-free graphs have light spanners. Specifically, they proved that every $K_h$-minor-free graph has a $(1+\epsilon)$-spanner of lightness $O_{\epsilon}(h \sqrt{\log h})$, hence constant when $h$ and $\epsilon$ are regarded as constants. We extend this result by showing that a more expressive size/stretch tradeoff is available. Specifically: for any positive integer $k$, every $n$-node, $K_h$-minor-free graph has a $(2k-1)$-spanner with sparsity \[O\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h\right),\] and a $(1+\epsilon)(2k-1)$-spanner with lightness \[O_{\epsilon}\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h \right).\] We further prove that this exponent $\frac{2}{k+1}$ is best possible, assuming the girth conjecture. At a technical level, our proofs leverage the recent improvements by Postle (2020) to the remarkable density increment theorem for minor-free graphs.

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arXiv2025

Light Edge Fault Tolerant Graph Spanners

There has recently been significant interest in fault tolerant spanners, which are spanners that still maintain their stretch guarantees after some nodes or edges fail. This work has culminated in an almost complete understanding of the three-way tradeoff between stretch, sparsity, and number of faults tolerated. However, despite some progress in metric settings, there have been no results to date on the tradeoff in general graphs between stretch, lightness, and number of faults tolerated. We initiate the study of light edge fault tolerant (EFT) graph spanners, obtaining the first such results. First, we observe that lightness can be unbounded if we use the traditional definition (normalizing by the MST). We then argue that a natural definition of fault-tolerant lightness is to instead normalize by a min-weight fault tolerant connectivity preserver; essentially, a fault-tolerant version of the MST. However, even with this, we show that it is still not generally possible to construct $f$-EFT spanners whose weight compares reasonably to the weight of a min-weight $f$-EFT connectivity preserver. In light of this lower bound, it is natural to then consider bicriteria notions of lightness, where we compare the weight of an $f$-EFT spanner to a min-weight $(f' > f)$-EFT connectivity preserver. The most interesting question is to determine the minimum value of $f'$ that allows for reasonable lightness upper bounds. Our main result is a precise answer to this question: $f' = 2f$. In particular, we show that the lightness can be untenably large (roughly $n/k$ for a $k$-spanner) if one normalizes by the min-weight $(2f-1)$-EFT connectivity preserver. But if one normalizes by the min-weight $2f$-EFT connectivity preserver, then we show that the lightness is bounded by just $O(f^{1/2})$ times the non-fault tolerant lightness (roughly $n^{1/k}$, for a $(1+\epsilon)(2k-1)$-spanner).

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arXiv2024

Improved Online Reachability Preservers

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 $|S|=:\sigma$, there is a construction that guarantees a maximum preserver size of $$|E(H)| \le O\left( (np\sigma)^{1/2} + n\right).$$ 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 $O( \sigma n)$ (folklore). All of these constructions are polynomial time, deterministic, and they do not require knowledge of the values of $p, \sigma$, 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 $O(n^{3/5 + \varepsilon})$.

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arXiv2024

A Lower Bound for Light Spanners in General Graphs

A recent upper bound by Le and Solomon [STOC '23] has established that every $n$-node graph has a $(1+\varepsilon)(2k-1)$-spanner with lightness $O(\varepsilon^{-1} n^{1/k})$. This bound is optimal up to its dependence on $\varepsilon$; the remaining open problem is whether this dependence can be improved or perhaps even removed entirely. We show that the $\varepsilon$-dependence cannot in fact be completely removed. For constant $k$ and for $\varepsilon:= \Theta(n^{-\frac{1}{2k-1}})$, we show a lower bound on lightness of $$\Omega\left( \varepsilon^{-1/k} n^{1/k} \right).$$ For example, this implies that there are graphs for which any $3$-spanner has lightness $\Omega(n^{2/3})$, improving on the previous lower bound of $\Omega(n^{1/2})$. An unusual feature of our lower bound is that it is conditional on the girth conjecture with parameter $k-1$ rather than $k$. We additionally show that this implies certain technical limitations to improving our lower bound further. In particular, under the same conditional, generalizing our lower bound to all $\varepsilon$ \emph{or} obtaining an optimal $\varepsilon$-dependence are as hard as proving the girth conjecture for all constant $k$.

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arXiv2024

Additive Spanner Lower Bounds with Optimal Inner Graph Structure

We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the first two steps of a particular three-step research program, introduced in prior work and overviewed here, aimed at producing tight bounds for the problem by aligning aspects of the upper and lower bound constructions. More specifically, we develop techniques that enable the use of inner graphs in the lower bound framework whose technical properties are provably tight with the corresponding assumptions made in the upper bounds. As an additional application of our techniques, we improve the corresponding lower bound for $O(n)$-size additive emulators to $+\Omega(n^{1/14})$.

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arXiv2024

The Discrepancy of Shortest Paths

The hereditary discrepancy of a set system is a certain quantitative measure of the pseudorandom properties of the system. Roughly, hereditary discrepancy measures how well one can $2$-color the elements of the system so that each set contains approximately the same number of elements of each color. Hereditary discrepancy has well-studied applications e.g. in communication complexity and derandomization. More recently, the hereditary discrepancy of set systems of shortest paths has found applications in differential privacy [Chen et al.~SODA 23]. The contribution of this paper is to improve the upper and lower bounds on the hereditary discrepancy of set systems of unique shortest paths in graphs. In particular, we show that any system of unique shortest paths in an undirected weighted graph has hereditary discrepancy $\widetilde{O}(n^{1/4})$, and we construct lower bound examples demonstrating that this bound is tight up to hidden $\text{polylog } n$ factors. Our lower bounds apply even in the planar and bipartite settings, and they improve on a previous lower bound of $\Omega(n^{1/6})$ obtained by applying the trace bound of Chazelle and Lvov [SoCG'00] to a classical point-line system of Erd\H{o}s. As applications, we improve the lower bound on the additive error for differentially-private all pairs shortest distances from $\Omega(n^{1/6})$ [Chen et al.~SODA 23] to $\Omega(n^{1/4})$, and we improve the lower bound on additive error for the differentially-private all sets range queries problem to $\Omega(n^{1/4})$, which is tight up to hidden $\text{polylog } n$ factors [Deng et al.~WADS 23].

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arXiv2023

Improved Shortest Path Restoration Lemmas for Multiple Edge Failures: Trade-offs Between Fault-tolerance and Subpaths

The restoration lemma is a classic result by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [PODC '01], which relates the structure of shortest paths in a graph $G$ before and after some edges in the graph fail. Their work shows that, after one edge failure, any replacement shortest path avoiding this failing edge can be partitioned into two pre-failure shortest paths. More generally, this implies an additive tradeoff between fault tolerance and subpath count: for any $f, k$, we can partition any $f$-edge-failure replacement shortest path into $k+1$ subpaths which are each an $(f-k)$-edge-failure replacement shortest path. This generalized result has found applications in routing, graph algorithms, fault tolerant network design, and more. Our main result improves this to a multiplicative tradeoff between fault tolerance and subpath count. We show that for all $f, k$, any $f$-edge-failure replacement path can be partitioned into $O(k)$ subpaths that are each an $(f/k)$-edge-failure replacement path. We also show an asymptotically matching lower bound. In particular, our results imply that the original restoration lemma is exactly tight in the case $k=1$, but can be significantly improved for larger $k$. We also show an extension of this result to weighted input graphs, and we give efficient algorithms that compute path decompositions satisfying our improved restoration lemmas.

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arXiv2023

Fault-Tolerant Spanners against Bounded-Degree Edge Failures: Linearly More Faults, Almost For Free

We study a new and stronger notion of fault-tolerant graph structures whose size bounds depend on the degree of the failing edge set, rather than the total number of faults. For a subset of faulty edges $F \subseteq G$, the faulty-degree $\deg(F)$ is the largest number of faults in $F$ incident to any given vertex. We design new fault-tolerant structures with size comparable to previous constructions, but which tolerate every fault set of small faulty-degree $\deg(F)$, rather than only fault sets of small size $|F|$. Our main results are: - New FT-Certificates: For every $n$-vertex graph $G$ and degree threshold $f$, one can compute a connectivity certificate $H \subseteq G$ with $|E(H)| = \widetilde{O}(fn)$ edges that has the following guarantee: for any edge set $F$ with faulty-degree $\deg(F)\leq f$ and every vertex pair $u,v$, it holds that $u$ and $v$ are connected in $H \setminus F$ iff they are connected in $G \setminus F$. This bound on $|E(H)|$ is nearly tight. Since our certificates handle some fault sets of size up to $|F|=O(fn)$, prior work did not imply any nontrivial upper bound for this problem, even when $f=1$. - New FT-Spanners: We show that every $n$-vertex graph $G$ admits a $(2k-1)$-spanner $H$ with $|E(H)| = O_k(f^{1-1/k} n^{1+1/k})$ edges, which tolerates any fault set $F$ of faulty-degree at most $f$. This bound on $|E(H)|$ optimal up to its hidden dependence on $k$, and it is close to the bound of $O_k(|F|^{1/2} n^{1+1/k} + |F|n)$ that is known for the case where the total number of faults is $|F|$ [Bodwin, Dinitz, Robelle SODA '22]. Our proof of this theorem is non-constructive, but by following a proof strategy of Dinitz and Robelle [PODC '20], we show that the runtime can be made polynomial by paying an additional $\text{polylog } n$ factor in spanner size.

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arXiv2023

Spanning Adjacency Oracles in Sublinear Time

Suppose we are given an $n$-node, $m$-edge input graph $G$, and the goal is to compute a spanning subgraph $H$ on $O(n)$ edges. This can be achieved in linear $O(m + n)$ time via breadth-first search. But can we hope for \emph{sublinear} runtime in some range of parameters? If the goal is to return $H$ as an adjacency list, there are simple lower bounds showing that $\Omega(m + n)$ runtime is necessary. If the goal is to return $H$ as an adjacency matrix, then we need $\Omega(n^2)$ time just to write down the entries of the output matrix. However, we show that neither of these lower bounds still apply if instead the goal is to return $H$ as an \emph{implicit} adjacency matrix, which we call an \emph{adjacency oracle}. An adjacency oracle is a data structure that gives a user the illusion that an adjacency matrix has been computed: it accepts edge queries $(u, v)$, and it returns in near-constant time a bit indicating whether $(u, v) \in E(H)$. Our main result is that one can construct an adjacency oracle for a spanning subgraph on at most $(1+\varepsilon)n$ edges, in $\tilde{O}(n \varepsilon^{-1})$ time, and that this construction time is near-optimal. Additional results include constructions of adjacency oracles for $k$-connectivity certificates and spanners, which are similarly sublinear on dense-enough input graphs. Our adjacency oracles are closely related to Local Computation Algorithms (LCAs) for graph sparsifiers; they can be viewed as LCAs with some computation moved to a preprocessing step, in order to speed up queries. Our oracles imply the first Local algorithm for computing sparse spanning subgraphs of general input graphs in $\tilde{O}(n)$ query time, which works by constructing our adjacency oracle, querying it once, and then throwing the rest of the oracle away. This addresses an open problem of Rubinfeld [CSR '17].

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arXiv2023

Are there graphs whose shortest path structure requires large edge weights?

The aspect ratio of a (positively) weighted graph $G$ is the ratio of its maximum edge weight to its minimum edge weight. Aspect ratio commonly arises as a complexity measure in graph algorithms, especially related to the computation of shortest paths. Popular paradigms are to interpolate between the settings of weighted and unweighted input graphs by incurring a dependence on aspect ratio, or by simply restricting attention to input graphs of low aspect ratio. This paper studies the effects of these paradigms, investigating whether graphs of low aspect ratio have more structured shortest paths than graphs in general. In particular, we raise the question of whether one can generally take a graph of large aspect ratio and reweight its edges, to obtain a graph with bounded aspect ratio while preserving the structure of its shortest paths. Our findings are: - Every weighted DAG on $n$ nodes has a shortest-paths preserving graph of aspect ratio $O(n)$. A simple lower bound shows that this is tight. - The previous result does not extend to general directed or undirected graphs; in fact, the answer turns out to be exponential in these settings. In particular, we construct directed and undirected $n$-node graphs for which any shortest-paths preserving graph has aspect ratio $2^{\Omega(n)}$. We also consider the approximate version of this problem, where the goal is for shortest paths in $H$ to correspond to approximate shortest paths in $G$. We show that our exponential lower bounds extend even to this setting. We also show that in a closely related model, where approximate shortest paths in $H$ must also correspond to approximate shortest paths in $G$, even DAGs require exponential aspect ratio.

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arXiv2023

An Alternate Proof of Near-Optimal Light Spanners

In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every $n$-node graph $G$ has a $(1+\varepsilon)(2k-1)$-spanner of lightness $O_{\varepsilon}(n^{1/k})$, and recent followup work by Le and Solomon [STOC '23] generalized the proof strategy and improved the dependence on $\varepsilon$. We give a new proof of this result, with the improved $\varepsilon$-dependence. Our proof is a direct analysis of the often-studied greedy spanner, and can be viewed as an extension of the folklore Moore bounds used to analyze spanner sparsity.

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arXiv2023

Folklore Sampling is Optimal for Exact Hopsets: Confirming the $\sqrt{n}$ 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 $p \in [1, n^2]$.

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arXiv2022

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.

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