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Greg Bodwin

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17 papers
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17

FOCS Conference 2023 Conference Paper

Bridge Girth: A Unifying Notion in Network Design

  • Greg Bodwin
  • Gary Hoppenworth
  • Ohad Trabelsi

A classic 1993 paper by Althöfer 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: •We write the maximum possible size of an n-node, p-path system with bridge girth $\gt k$ 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 polynomial 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 $\beta^{*}(n, p, \infty)$. •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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FOCS Conference 2023 Conference Paper

Folklore Sampling is Optimal for Exact Hopsets: Confirming the √n Barrier

  • Greg Bodwin
  • Gary Hoppenworth

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 rather than distances. 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 algorithm for shortcut sets and for $(1+\varepsilon)$-approximate hopsets. For either object, it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1 / 3})$, improving over the folklore diameter bound of $\widetilde{O}(n^{1 / 2})$. The only setting in which 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 $\Omega(n^{1 / 3})$ 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 parameters $p \in[1, n^{2}]$.

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FOCS Conference 2022 Conference Paper

New Additive Spanner Lower Bounds by an Unlayered Obstacle Product

  • Greg Bodwin
  • Gary Hoppenworth

For an input graph G, an additive spanner is a sparse subgraph H whose shortest paths match those of G up to small additive error. We prove two new lower bounds in the area of additive spanners: •We construct n-node graphs G for which any spanner on $O(n)$ edges must increase a pairwise distance by $+\Omega(n^{1/7})$. This improves on a recent lower bound of $+\Omega(n^{1/10. 5})$ by Lu, Wein, Vassilevska Williams, and Xu [SODA 22]. •A classic result by Coppersmith and Elkin [SODA 05] proves that for any n-node graph G and set of $p=O(n^{1/2})$ demand pairs, one can exactly preserve all pairwise distances among demand pairs using a spanner on $O(n)$ edges. They also provided a lower bound construction, establishing that that this range $p=O(n^{1/2})$ cannot be improved. We strengthen this lower bound by proving that, for any constant k, this range of p is still unimprovable even if the spanner is allowed $+k$ additive error among the demand pairs. This negatively resolves an open question asked by Coppersmith and Elkin [SODA 05] and again by Cygan, Grandoni, and Kavitha [STACS 13] and Abboud and Bodwin [SODA 16]. At a technical level, our lower bounds are obtained by an improvement to the entire obstacle product framework used to compose “inner” and outer” graphs into lower bound instances. In particular, we develop a new strategy for analysis that allows certain non-layered graphs to be used in the product, and we use this freedom to design better inner and outer graphs that lead to our new lower bounds.

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STOC Conference 2016 Conference Paper

The 4/3 additive spanner exponent is tight

  • Amir Abboud
  • Greg Bodwin

A spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph. It is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation, so long as distance error is measured multiplicatively . A central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of additive error. That is, is it true that for all ε>0, there is a constant k ε such that every graph has a spanner on O ( n 1+ε ) edges that preserves its pairwise distances up to + k ε ? Previous lower bounds are consistent with a positive resolution to this question, while previous upper bounds exhibit the beginning of a tradeoff curve: all graphs have +2 spanners on O ( n 3/2 ) edges, +4 spanners on Õ( n 7/5 ) edges, and +6 spanners on O ( n 4/3 ) edges. However, progress has mysteriously halted at the n 4/3 bound, and despite significant effort from the community, the question has remained open for all 0 < ε < 1/3. Our main result is a surprising negative resolution of the open question, even in a highly generalized setting. We show a new information theoretic incompressibility bound: there is no function that compresses graphs into O ( n 4/3 − ε ) bits so that distance information can be recovered within + n o (1) error. As a special case of our theorem, we get a tight lower bound on the sparsity of additive spanners: the +6 spanner on O ( n 4/3 ) edges cannot be improved in the exponent, even if any subpolynomial amount of additive error is allowed. Our theorem implies new lower bounds for related objects as well; for example, the twenty-year-old +4 emulator on O ( n 4/3 ) edges also cannot be improved in the exponent unless the error allowance is polynomial. Central to our construction is a new type of graph product, which we call the Obstacle Product . Intuitively, it takes two graphs G , H and produces a new graph G H whose shortest paths structure looks locally like H but globally like G .

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