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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

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^{Ω(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.

Aaron Bernstein, Greg Bodwin, and Nicole Wein. Are There Graphs Whose Shortest Path Structure Requires Large Edge Weights?. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bernstein_et_al:LIPIcs.ITCS.2024.12, author = {Bernstein, Aaron and Bodwin, Greg and Wein, Nicole}, title = {{Are There Graphs Whose Shortest Path Structure Requires Large Edge Weights?}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.12}, URN = {urn:nbn:de:0030-drops-195405}, doi = {10.4230/LIPIcs.ITCS.2024.12}, annote = {Keywords: shortest paths, graph theory, weighted graphs} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

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 sublinear runtime in some range of parameters - for example, perhaps O(n^{1.9}) worst-case runtime, even when the input graph has n² edges?
If the goal is to return H as an adjacency list, there are simple lower bounds showing that Ω(m + n) runtime is necessary. If the goal is to return H as an adjacency matrix, then we need Ω(n²) 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 implicit adjacency matrix, which we call an 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 or not (u, v) ∈ E(H).
Our main result is that, for any 0 < ε < 1, one can construct an adjacency oracle for a spanning subgraph on at most (1+ε)n edges, in Õ(n ε^{-1}) time (hence sublinear time on input graphs with m ≫ n edges), 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 LCAs for computing sparse spanning subgraphs of general input graphs in Õ(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].

Greg Bodwin and Henry Fleischmann. Spanning Adjacency Oracles in Sublinear Time. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 19:1-19:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bodwin_et_al:LIPIcs.ITCS.2024.19, author = {Bodwin, Greg and Fleischmann, Henry}, title = {{Spanning Adjacency Oracles in Sublinear Time}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {19:1--19:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.19}, URN = {urn:nbn:de:0030-drops-195475}, doi = {10.4230/LIPIcs.ITCS.2024.19}, annote = {Keywords: Graph algorithms, Sublinear algorithms, Data structures, Graph theory} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

A t-emulator of a graph G is a graph H that approximates its pairwise shortest path distances up to multiplicative t error. We study fault tolerant t-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior.
In particular, our main result is that, for (2k-1)-emulators with k odd, we can tolerate a polynomial number of edge faults for free. For example: for any n-node input graph, we construct a 5-emulator (k = 3) on O(n^{4/3}) edges that is robust to f = O(n^{2/9}) edge faults. It is well known that Ω(n^{4/3}) edges are necessary even if the 5-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd k, and whether a similar phenomenon can be proved for even k.

Greg Bodwin, Michael Dinitz, and Yasamin Nazari. Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 20:1-20:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodwin_et_al:LIPIcs.ITCS.2023.20, author = {Bodwin, Greg and Dinitz, Michael and Nazari, Yasamin}, title = {{Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {20:1--20:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.20}, URN = {urn:nbn:de:0030-drops-175231}, doi = {10.4230/LIPIcs.ITCS.2023.20}, annote = {Keywords: Emulators, Fault Tolerance, Girth Conjecture} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

In competitive games, it is common to assign each player a real number rating signifying their skill level. A rating system is a procedure by which player ratings are adjusted upwards each time they win, or downwards each time they lose.
Many matchmaking systems give players some control over their opponent’s rating; for example, a player might be able to selectively initiate games against opponents whose ratings are publicly visible, or abort a game without penalty before it begins but after glimpsing their opponent’s rating. It is natural to ask whether one can design a rating system that does not incentivize a rating-maximizing player to act strategically, seeking games against opponents of one rating over another. We show the following:
- The full version of this "opponent indifference" property is unfortunately too strong to be feasible. Although it is satisfied by some rating systems, these systems lack certain desirable expressiveness properties, suggesting that they are not suitable to capture most games of interest.
- However, there is a natural relaxation, roughly requiring indifference between any two opponents who are both "reasonably evenly matched" with the choosing player. We prove that this relaxed variant of opponent indifference, which we call P opponent indifference, is viable. In fact, a certain strong version of P opponent indifference precisely characterizes the rating system Sonas, which was originally proposed for its empirical predictive accuracy on the outcomes of high-level chess games.

Greg Bodwin and Forest Zhang. Opponent Indifference in Rating Systems: A Theoretical Case for Sonas. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodwin_et_al:LIPIcs.ITCS.2023.21, author = {Bodwin, Greg and Zhang, Forest}, title = {{Opponent Indifference in Rating Systems: A Theoretical Case for Sonas}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {21:1--21:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.21}, URN = {urn:nbn:de:0030-drops-175242}, doi = {10.4230/LIPIcs.ITCS.2023.21}, annote = {Keywords: Rating systems, opponent indifference, incentive compatibility, mechanism design, game theory} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

A k-spanner of a graph G is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of k, and a k-emulator is similar but not required to be a subgraph of G. A classic theorem by Althöfer et al. [Disc. Comp. Geom. '93] and Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners.
We complement our emulator upper bound with a lower bound construction that is essentially tight (within log n factors of the upper bound) when the stretch is 2k-1 and k is either a fixed odd integer or 2. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.

Greg Bodwin, Michael Dinitz, and Yasamin Nazari. Vertex Fault-Tolerant Emulators. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bodwin_et_al:LIPIcs.ITCS.2022.25, author = {Bodwin, Greg and Dinitz, Michael and Nazari, Yasamin}, title = {{Vertex Fault-Tolerant Emulators}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {25:1--25:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.25}, URN = {urn:nbn:de:0030-drops-156217}, doi = {10.4230/LIPIcs.ITCS.2022.25}, annote = {Keywords: Emulators, Fault Tolerance, Girth Conjecture} }

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**Published in:** LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Given a graph G = (V,E), a subgraph H is an additive +β spanner if dist_H(u,v) ≤ dist_G(u,v) + β for all u, v ∈ V. A pairwise spanner is a spanner for which the above inequality is only required to hold for specific pairs P ⊆ V × V given on input; when the pairs have the structure P = S × S for some S ⊆ V, it is called a subsetwise spanner. Additive spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs.
In this paper, we consider a multi-level version of the subsetwise additive spanner in weighted graphs motivated by multi-level network design and visualization, where the vertices in S possess varying level, priority, or quality of service (QoS) requirements. The goal is to compute a nested sequence of spanners with the minimum total number of edges. We first generalize the +2 subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several existing algorithms by [Ahmed et al., 2020] for weighted additive spanners, both in terms of runtime and sparsity of the output spanner, when applied as a subroutine to multi-level problem.
We provide an experimental evaluation on graphs using several different random graph generators and show that these spanner algorithms typically achieve much better guarantees in terms of sparsity and additive error compared with the theoretical maximum. By analyzing our experimental results, we additionally developed a new technique of changing a certain initialization parameter which provides better spanners in practice at the expense of a small increase in running time.

Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence. Multi-Level Weighted Additive Spanners. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ahmed_et_al:LIPIcs.SEA.2021.16, author = {Ahmed, Reyan and Bodwin, Greg and Sahneh, Faryad Darabi and Hamm, Keaton and Kobourov, Stephen and Spence, Richard}, title = {{Multi-Level Weighted Additive Spanners}}, booktitle = {19th International Symposium on Experimental Algorithms (SEA 2021)}, pages = {16:1--16:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-185-6}, ISSN = {1868-8969}, year = {2021}, volume = {190}, editor = {Coudert, David and Natale, Emanuele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.16}, URN = {urn:nbn:de:0030-drops-137885}, doi = {10.4230/LIPIcs.SEA.2021.16}, annote = {Keywords: multi-level, graph spanner, approximation algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Fault tolerant distance preservers are sparse subgraphs that preserve distances between given pairs of nodes under edge or vertex failures. In this paper, we present the first non-trivial constructions of subset distance preservers, which preserve all distances among a subset of nodes S, that can handle either an edge or a vertex fault.
- For an n-vertex undirected weighted graph or weighted DAG G = (V,E) and S ⊆ V, we present a construction of a subset preserver with Õ(|S|n) edges that is resilient to a single fault. In the single pair case (|S| = 2), the bound improves to O(n). We further provide a nearly-matching lower bound of Ω(|S|n) in either setting, and we show that the same lower bound holds conditionally even if attention is restricted to unweighted graphs.
- For an n-vertex directed unweighted graph G = (V,E) and r ∈ V, S ⊆ V, we present a construction of a preserver of distances in {r} × S with Õ(n^{4/3} |S|^{5/6}) edges that is resilient to a single fault. In the case |S| = 1 the bound improves to O(n^{4/3}), and for this case we provide another matching conditional lower bound.
- For an n-vertex directed weighted graph G = (V, E) and r ∈ V, S ⊆ V, we present a construction of a preserver of distances in {r} × S with Õ(n^{3/2} |S|^{3/4}) edges that is resilient to a single vertex fault. (It was proved in [Greg Bodwin et al., 2017] that the bound improves to O(n^{3/2}) when |S| = 1, and that this is conditionally tight.)

Greg Bodwin, Keerti Choudhary, Merav Parter, and Noa Shahar. New Fault Tolerant Subset Preservers. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bodwin_et_al:LIPIcs.ICALP.2020.15, author = {Bodwin, Greg and Choudhary, Keerti and Parter, Merav and Shahar, Noa}, title = {{New Fault Tolerant Subset Preservers}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {15:1--15:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.15}, URN = {urn:nbn:de:0030-drops-124222}, doi = {10.4230/LIPIcs.ICALP.2020.15}, annote = {Keywords: Subset Preservers, Distances, Fault-tolerance} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists? We prove that this problem is PSPACE-Complete already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments in the current literature.

Greg Bodwin and Ofer Grossman. Strategy-Stealing Is Non-Constructive. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bodwin_et_al:LIPIcs.ITCS.2020.21, author = {Bodwin, Greg and Grossman, Ofer}, title = {{Strategy-Stealing Is Non-Constructive}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {21:1--21:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.21}, URN = {urn:nbn:de:0030-drops-117069}, doi = {10.4230/LIPIcs.ITCS.2020.21}, annote = {Keywords: PSPACE-hard, Hex, Combinatorial Game Theory} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

This paper develops a recent line of economic theory seeking to understand public goods economies using methods of topological analysis. Our first main result is a very clean characterization of the economy's core (the standard solution concept in public goods). Specifically, we prove that a point is in the core iff it is Pareto efficient, individually rational, and the set of points it dominates is path connected.
While this structural theorem has a few interesting implications in economic theory, the main focus of the second part of this paper is on a particular algorithmic application that demonstrates its utility. Since the 1960s, economists have looked for an efficient computational process that decides whether or not a given point is in the core. All known algorithms so far run in exponential time (except in some artificially restricted settings). By heavily exploiting our new structure, we propose a new algorithm for testing core membership whose computational bottleneck is the solution of O(n) convex optimization problems on the utility function governing the economy. It is fairly natural to assume that convex optimization should be feasible, as it is needed even for very basic economic computational tasks such as testing Pareto efficiency. Nevertheless, even without this assumption, our work implies for the first time that core membership can be efficiently tested on (e.g.) utility functions that admit ``nice'' analytic expressions, or that appropriately defined epsilon-approximate versions of the problem are tractable (by using modern black-box epsilon-approximate convex optimization algorithms).

Greg Bodwin. Testing Core Membership in Public Goods Economies. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bodwin:LIPIcs.ICALP.2017.45, author = {Bodwin, Greg}, title = {{Testing Core Membership in Public Goods Economies}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.45}, URN = {urn:nbn:de:0030-drops-74910}, doi = {10.4230/LIPIcs.ICALP.2017.45}, annote = {Keywords: Algorithmic Game Theory, Economics, Algorithms, Public Goods, Coalitional Stability} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain replacement paths around the faulted set. Unfortunately, the complexity of the interaction between replacement paths blows up significantly, even from 1 to 2 faults, and the structure of optimal preservers and spanners is poorly understood. In particular, no nontrivial bounds for preservers and additive spanners are known when the number of faults is bigger than 2.
Even the answer to the following innocent question is completely unknown: what is the worst-case size of a preserver for a single pair of nodes in the presence of f edge faults? There are no super-linear lower bounds, nor subquadratic upper bounds for f>2. In this paper we make substantial progress on this and other fundamental questions:
- We present the first truly sub-quadratic size fault-tolerant single-pair preserver in unweighted (possibly directed) graphs: for any n node graph and any fixed number f of faults, O~(fn^{2-1/2^f}) size suffices. Our result also generalizes to the single-source (all targets) case, and can be used to build new fault-tolerant additive spanners (for all pairs).
- The size of the above single-pair preserver grows to O(n^2) for increasing f. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size O(n^{2-eps}) for \eps>0, then the additive error has to be \Omega(eps f). This surprisingly matches known upper bounds in the literature.
- For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is Theta(n^2) for f > 1 in both directed and undirected graphs, while for f=1 the size is Theta(n) in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound.
Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure.

Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. Preserving Distances in Very Faulty Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bodwin_et_al:LIPIcs.ICALP.2017.73, author = {Bodwin, Greg and Grandoni, Fabrizio and Parter, Merav and Vassilevska Williams, Virginia}, title = {{Preserving Distances in Very Faulty Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {73:1--73:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.73}, URN = {urn:nbn:de:0030-drops-74906}, doi = {10.4230/LIPIcs.ICALP.2017.73}, annote = {Keywords: Fault Tolerance, shortest paths, replacement paths} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

An alpha-spanner of a graph G is a subgraph H such that H preserves all distances of G within a factor of alpha. In this paper, we give fully dynamic algorithms for maintaining a spanner H of a graph G undergoing edge insertions and deletions with worst-case guarantees on the running time after each update. In particular, our algorithms maintain:
- a 3-spanner with ~O(n^{1+1/2}) edges with worst-case update time ~O(n^{3/4}), or
- a 5-spanner with ~O(n^{1+1/3}) edges with worst-case update time ~O (n^{5/9}).
These size/stretch tradeoffs are best possible (up to logarithmic factors). They can be extended to the weighted setting at very minor cost. Our algorithms are randomized and correct with high probability against an oblivious adversary. We also further extend our techniques to construct a 5-spanner with suboptimal size/stretch tradeoff, but improved worst-case update time.
To the best of our knowledge, these are the first dynamic spanner algorithms with sublinear worst-case update time guarantees. Since it is known how to maintain a spanner using small amortized}but large worst-case update time [Baswana et al. SODA'08], obtaining algorithms with strong worst-case bounds, as presented in this paper, seems to be the next natural step for this problem.

Greg Bodwin and Sebastian Krinninger. Fully Dynamic Spanners with Worst-Case Update Time. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bodwin_et_al:LIPIcs.ESA.2016.17, author = {Bodwin, Greg and Krinninger, Sebastian}, title = {{Fully Dynamic Spanners with Worst-Case Update Time}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {17:1--17:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.17}, URN = {urn:nbn:de:0030-drops-63688}, doi = {10.4230/LIPIcs.ESA.2016.17}, annote = {Keywords: Dynamic graph algorithms, spanners} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Graph reconstruction algorithms seek to learn a hidden graph by repeatedly querying a black-box oracle for information about the graph structure. Perhaps the most well studied and applied version of the problem uses a distance oracle, which can report the shortest path distance between any pair of nodes.
We introduce and study the betweenness oracle, where bet(a, m, z) is true iff m lies on a shortest path between a and z. This oracle is strictly weaker than a distance oracle, in the sense that a betweenness query can be simulated by a constant number of distance queries, but not vice versa. Despite this, we are able to develop betweenness reconstruction algorithms that match the current state of the art for distance reconstruction, and even improve it for certain types of graphs. We obtain the following algorithms: (1) Reconstruction of general graphs in O(n^2) queries, (2) Reconstruction of degree-bounded graphs in ~O(n^{3/2}) queries, (3) Reconstruction of geodetic degree-bounded graphs in ~O(n) queries
In addition to being a fundamental graph theoretic problem with some natural applications, our new results shed light on some avenues for progress in the distance reconstruction problem.

Mikkel Abrahamsen, Greg Bodwin, Eva Rotenberg, and Morten Stöckel. Graph Reconstruction with a Betweenness Oracle. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abrahamsen_et_al:LIPIcs.STACS.2016.5, author = {Abrahamsen, Mikkel and Bodwin, Greg and Rotenberg, Eva and St\"{o}ckel, Morten}, title = {{Graph Reconstruction with a Betweenness Oracle}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.5}, URN = {urn:nbn:de:0030-drops-57068}, doi = {10.4230/LIPIcs.STACS.2016.5}, annote = {Keywords: graph reconstruction, bounded degree graphs, query complexity} }

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