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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

In STOC'95 [S. Arya et al., 1995] Arya et al. showed that any set of n points in ℝ^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter k with O(n α_k(n)) edges, for any k ≥ 2. The function α_k is the inverse of a certain Ackermann-style function, where α₀(n) = ⌈n/2⌉, α₁(n) = ⌈√n⌉, α₂(n) = ⌈log n⌉, α₃(n) = ⌈log log n⌉, α₄(n) = log^* n, α₅(n) = ⌊ 1/2 log^*n ⌋, …. Roughly speaking, for k ≥ 2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars.
Despite a large body of work on spanners of bounded hop-diameter, the fundamental question of whether this tradeoff between size and hop-diameter of Euclidean (1+ε)-spanners is optimal has remained open, even in one-dimensional spaces. Three lower bound tradeoffs are known:
- An optimal k versus Ω(n α_k(n)) by Alon and Schieber [N. Alon and B. Schieber, 1987], but it applies to stretch 1 (not 1+ε).
- A suboptimal k versus Ω(nα_{2k+6}(n)) by Chan and Gupta [H. T.-H. Chan and A. Gupta, 2006].
- A suboptimal k versus Ω(n/(2^{6⌊k/2⌋}) α_k(n)) by Le et al. [Le et al., 2022]. This paper establishes the optimal k versus Ω(n α_k(n)) lower bound tradeoff for stretch 1+ε, for any ε > 0, and for any k. An important conceptual contribution of this work is in achieving optimality by shaving off an extremely slowly growing term, namely 2^{6⌊k/2⌋} for k ≤ O(α(n)); such a fine-grained optimization (that achieves optimality) is very rare in the literature.
To shave off the 2^{6⌊k/2⌋} term from the previous bound of Le et al., our argument has to drill much deeper. In particular, we propose a new way of analyzing recurrences that involve inverse-Ackermann style functions, and our key technical contribution is in presenting the first explicit construction of concave versions of these functions. An important advantage of our approach over previous ones is its robustness: While all previous lower bounds are applicable only to restricted 1-dimensional point sets, ours applies even to random point sets in constant-dimensional spaces.

Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{le_et_al:LIPIcs.SoCG.2023.47, author = {Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay}, title = {{Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {47:1--47:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.47}, URN = {urn:nbn:de:0030-drops-178976}, doi = {10.4230/LIPIcs.SoCG.2023.47}, annote = {Keywords: Euclidean spanners, Ackermann functions, convex functions, hop-diameter} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(n α_k(n)) edges, for any k≥2. The function α_k is the inverse of a certain Ackermann-style function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α₀(n)=⌈n/2⌉, α₁(n)=⌈√n⌉, α₂(n)=⌈log n⌉, α₃(n)=⌈log log n⌉, α₄(n)=log^* n, α₅(n)=⌊1/2 log^*n⌋, .... Roughly speaking, for k≥2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Also, α_{2α(n)+4}(n)≤4, where α(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function.
Whether or not this tradeoff is tight has remained open, even for the cases k=2 and k=3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ε>0, any (1+ε)-spanner for the uniform line metric with hop-diameter at most k must have at least Ω(n α_k(n)) edges.

Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 54:1-54:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{le_et_al:LIPIcs.SoCG.2022.54, author = {Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay}, title = {{Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {54:1--54:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.54}, URN = {urn:nbn:de:0030-drops-160629}, doi = {10.4230/LIPIcs.SoCG.2022.54}, annote = {Keywords: Euclidean spanners, hop-diameter, inverse-Ackermann, lower bounds, sparse spanners} }

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

The field of dynamic graph algorithms aims at achieving a thorough understanding of real-world networks whose topology evolves with time. Traditionally, the focus has been on the classic sequential, centralized setting where the main quality measure of an algorithm is its update time, i.e. the time needed to restore the solution after each update. While real-life networks are very often distributed across multiple machines, the fundamental question of finding efficient dynamic, distributed graph algorithms received little attention to date. The goal in this setting is to optimize both the round and message complexities incurred per update step, ideally achieving a message complexity that matches the centralized update time in O(1) (perhaps amortized) rounds.
Toward initiating a systematic study of dynamic, distributed algorithms, we study some of the most central symmetry-breaking problems: maximal independent set (MIS), maximal matching/(approx-) maximum cardinality matching (MM/MCM), and (Δ + 1)-vertex coloring. This paper focuses on dynamic, distributed algorithms that are deterministic, and in particular - robust against an adaptive adversary. Most of our focus is on our MIS algorithm, which achieves O (m^{2/3}log² n) amortized messages in O(log² n) amortized rounds in the Congest model. Notably, the amortized message complexity of our algorithm matches the amortized update time of the best-known deterministic centralized MIS algorithm by Gupta and Khan [SOSA'21] up to a polylog n factor. The previous best deterministic distributed MIS algorithm, by Assadi et al. [STOC'18], uses O(m^{3/4}) amortized messages in O(1) amortized rounds, i.e., we achieve a polynomial improvement in the message complexity by a polylog n increase to the round complexity; moreover, the algorithm of Assadi et al. makes an implicit assumption that the network is connected at all times, which seems excessively strong when it comes to dynamic networks. Using techniques similar to the ones we developed for our MIS algorithm, we also provide deterministic algorithms for MM, approximate MCM and (Δ + 1)-vertex coloring whose message complexities match or nearly match the update times of the best centralized algorithms, while having either constant or polylog(n) round complexities.

Shiri Antaki, Quanquan C. Liu, and Shay Solomon. Near-Optimal Distributed Implementations of Dynamic Algorithms for Symmetry Breaking Problems. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 7:1-7:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{antaki_et_al:LIPIcs.ITCS.2022.7, author = {Antaki, Shiri and Liu, Quanquan C. and Solomon, Shay}, title = {{Near-Optimal Distributed Implementations of Dynamic Algorithms for Symmetry Breaking Problems}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {7:1--7:25}, 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.2022.7}, URN = {urn:nbn:de:0030-drops-156039}, doi = {10.4230/LIPIcs.ITCS.2022.7}, annote = {Keywords: dynamic graph algorithms, distributed algorithms, symmetry breaking problems, maximal independent set, matching, coloring} }

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

Dynamic graph matching algorithms have been extensively studied, but mostly under edge updates. This paper concerns dynamic matching algorithms under vertex updates, where in each update step a single vertex is either inserted or deleted along with its incident edges.
A basic setting arising in online algorithms and studied by Bosek et al. [FOCS'14] and Bernstein et al. [SODA'18] is that of dynamic approximate maximum cardinality matching (MCM) in bipartite graphs in which one side is fixed and vertices on the other side either arrive or depart via vertex updates. In the BASIC-incremental setting, vertices only arrive, while in the BASIC-decremental setting vertices only depart. When vertices can both arrive and depart, we have the BASIC-dynamic setting. In this paper we also consider the setting in which both sides of the bipartite graph are dynamic. We call this the MEDIUM-dynamic setting, and MEDIUM-decremental is the restriction when vertices can only depart. The GENERAL-dynamic setting is when the graph is not necessarily bipartite and the vertices can both depart and arrive.
Denote by K the total number of edges inserted and deleted to and from the graph throughout the entire update sequence. A well-studied measure, the recourse of a dynamic matching algorithm is the number of changes made to the matching per step. We largely focus on Maximal Matching (MM) which is a 2-approximation to the MCM. Our main results are as follows.
- In the BASIC-dynamic setting, there is a straightforward algorithm for maintaining a MM, with a total runtime of O(K) and constant worst-case recourse. In fact, this algorithm never removes an edge from the matching; we refer to such an algorithm as irrevocable.
- For the MEDIUM-dynamic setting we give a strong conditional lower bound that even holds in the MEDIUM-decremental setting: if for any fixed η > 0, there is an irrevocable decremental MM algorithm with a total runtime of O(K ⋅ n^{1-η}), this would refute the OMv conjecture; a similar (but weaker) hardness result can be achieved via a reduction from the Triangle Detection conjecture.
- Next, we consider the GENERAL-dynamic setting, and design an MM algorithm with a total runtime of O(K) and constant worst-case recourse. We achieve this result via a 1-revocable algorithm, which may remove just one edge per update step. As argued above, an irrevocable algorithm with such a runtime is not likely to exist.
- Finally, back to the BASIC-dynamic setting, we present an algorithm with a total runtime of O(K), which provides an (e/(e-1))-approximation to the MCM.
To this end, we build on the classic "ranking" online algorithm by Karp et al. [STOC'90]. Beyond the results, our work draws connections between the areas of dynamic graph algorithms and online algorithms, and it proposes several open questions that seem to be overlooked thus far.

Hung Le, Lazar Milenković, Shay Solomon, and Virginia Vassilevska Williams. Dynamic Matching Algorithms Under Vertex Updates. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 96:1-96:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{le_et_al:LIPIcs.ITCS.2022.96, author = {Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Vassilevska Williams, Virginia}, title = {{Dynamic Matching Algorithms Under Vertex Updates}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {96:1--96:24}, 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.2022.96}, URN = {urn:nbn:de:0030-drops-156923}, doi = {10.4230/LIPIcs.ITCS.2022.96}, annote = {Keywords: maximal matching, approximate matching, dynamic algorithm, vertex updates} }

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**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

We revisit the minimum dominating set problem on graphs with arboricity bounded by α. In the (standard) centralized setting, Bansal and Umboh [Bansal and Umboh, 2017] gave an O(α)-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [Koufogiannakis and Young, 2014; Young, 2014; Allen-Zhu and Orecchia, 2019; Quanrud, 2020]. Moreover, [Bansal and Umboh, 2017] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [Christoph Lenzen and Roger Wattenhofer, 2010], and Jones et al. [Jones et al., 2013], achieve an approximation factor of O(α²) in linear time.
There is a similar situation in the distributed setting: While there is an O(log² n)-round LP-based O(α)-approximation algorithm implied in [Kuhn et al., 2006], the best non-LP-based algorithm by Lenzen and Wattenhofer [Christoph Lenzen and Roger Wattenhofer, 2010] is an implementation of their centralized algorithm, providing an O(α²)-approximation within O(log n) rounds.
We address the questions of whether one can achieve an O(α)-approximation algorithm that is elementary, i.e., not based on any LP-based methods, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. Our contribution is two-fold:
1) In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an O(α)-approximation in linear time. The previous state-of-the-art O(α)-approximation algorithms are (1) LP-based, (2) more complicated, and (3) have super-linear running time.
2) Based on our centralized algorithm, we design a distributed combinatorial O(α)-approximation algorithm in the CONGEST model that runs in O(αlog n) rounds with high probability. Not only does this result provide the first nontrivial non-LP-based distributed o(α²)-approximation algorithm for this problem, it also outperforms the best LP-based distributed algorithm for a wide range of parameters.

Adir Morgan, Shay Solomon, and Nicole Wein. Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{morgan_et_al:LIPIcs.DISC.2021.33, author = {Morgan, Adir and Solomon, Shay and Wein, Nicole}, title = {{Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {33:1--33:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.33}, URN = {urn:nbn:de:0030-drops-148353}, doi = {10.4230/LIPIcs.DISC.2021.33}, annote = {Keywords: Graph Algorithms, Dominating Set, Bounded Arboricity, Linear time algorithms} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

In the (fully) dynamic set cover problem, we have a collection of m sets from a universe of size n that undergo element insertions and deletions; the goal is to maintain an approximate set cover of the universe after each update. We give an O(f²) update time algorithm for this problem that achieves an f-approximation, where f is the maximum number of sets that an element belongs to; under the unique games conjecture, this approximation is best possible for any fixed f. This is the first algorithm for dynamic set cover with approximation ratio that exactly matches f (as opposed to almost f in prior work), as well as the first one with runtime independent of n,m (for any approximation factor of o(f³)).
Prior to our work, the state-of-the-art algorithms for this problem were O(f²) update time algorithms of Gupta et al. [STOC'17] and Bhattacharya et al. [IPCO'17] with O(f³) approximation, and the recent algorithm of Bhattacharya {et al. } [FOCS'19] with O(f⋅log{n}/ε²) update time and (1+ε)⋅f approximation, improving the O(f²⋅log{n}/ε⁵) bound of Abboud et al. [STOC'19].
The key technical ingredient of our work is an algorithm for maintaining a maximal matching in a dynamic hypergraph of rank r - where each hyperedge has at most r vertices - that undergoes hyperedge insertions and deletions in O(r²) amortized update time; our algorithm is randomized, and the bound on the update time holds in expectation and with high probability. This result generalizes the maximal matching algorithm of Solomon [FOCS'16] with constant update time in ordinary graphs to hypergraphs, and is of independent merit; the previous state-of-the-art algorithms for set cover do not translate to (integral) matchings for hypergraphs, let alone a maximal one. Our quantitative result for the set cover problem is translated directly from this qualitative result for maximal matching using standard reductions.
An important advantage of our approach over the previous ones for approximation (1+ε)⋅f (by Abboud et al. [STOC'19] and Bhattacharya et al. [FOCS'19]) is that it is inherently local and can thus be distributed efficiently to achieve low amortized round and message complexities.

Sepehr Assadi and Shay Solomon. Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{assadi_et_al:LIPIcs.ESA.2021.8, author = {Assadi, Sepehr and Solomon, Shay}, title = {{Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.8}, URN = {urn:nbn:de:0030-drops-145899}, doi = {10.4230/LIPIcs.ESA.2021.8}, annote = {Keywords: dynamic graph algorithms, hypergraph, maximal matching, matching, set cover} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

The goal in reconfiguration problems is to compute a gradual transformation between two feasible solutions of a problem such that all intermediate solutions are also feasible. In the Matching Reconfiguration Problem (MRP), proposed in a pioneering work by Ito et al. from 2008, we are given a graph G and two matchings M and M', and we are asked whether there is a sequence of matchings in G starting with M and ending at M', each resulting from the previous one by either adding or deleting a single edge in G, without ever going through a matching of size < min{|M|,|M'|}-1. Ito et al. gave a polynomial time algorithm for the problem, which uses the Edmonds-Gallai decomposition.
In this paper we introduce a natural generalization of the MRP that depends on an integer parameter Δ ≥ 1: here we are allowed to make Δ changes to the current solution rather than 1 at each step of the {transformation procedure}. There is always a valid sequence of matchings transforming M to M' if Δ is sufficiently large, and naturally we would like to minimize Δ. We first devise an optimal transformation procedure for unweighted matching with Δ = 3, and then extend it to weighted matchings to achieve asymptotically optimal guarantees. The running time of these procedures is linear.
We further demonstrate the applicability of this generalized problem to dynamic graph matchings. In this area, the number of changes to the maintained matching per update step (the recourse bound) is an important quality measure. Nevertheless, the worst-case recourse bounds of almost all known dynamic matching algorithms are prohibitively large, much larger than the corresponding update times. We fill in this gap via a surprisingly simple black-box reduction: Any dynamic algorithm for maintaining a β-approximate maximum cardinality matching with update time T, for any β ≥ 1, T and ε > 0, can be transformed into an algorithm for maintaining a (β(1 +ε))-approximate maximum cardinality matching with update time T + O(1/ε) and worst-case recourse bound O(1/ε). This result generalizes for approximate maximum weight matching, where the update time and worst-case recourse bound grow from T + O(1/ε) and O(1/ε) to T + O(ψ/ε) and O(ψ/ε), respectively; ψ is the graph aspect-ratio. We complement this positive result by showing that, for β = 1+ε, the worst-case recourse bound of any algorithm produced by our reduction is optimal. As a corollary, several key dynamic approximate matching algorithms - with poor worst-case recourse bounds - are strengthened to achieve near-optimal worst-case recourse bounds with no loss in update time.

Noam Solomon and Shay Solomon. A Generalized Matching Reconfiguration Problem. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 57:1-57:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{solomon_et_al:LIPIcs.ITCS.2021.57, author = {Solomon, Noam and Solomon, Shay}, title = {{A Generalized Matching Reconfiguration Problem}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {57:1--57:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.57}, URN = {urn:nbn:de:0030-drops-135965}, doi = {10.4230/LIPIcs.ITCS.2021.57}, annote = {Keywords: Dynamic algorithms, graph matching, reconfiguration problem, recourse bound} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

The FOCS'19 paper of Le and Solomon [Hung Le and Shay Solomon, 2019], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1+ε)-spanner in ℝ^d is Õ(ε^{-d}) for any d = O(1) and any ε = Ω(n^{-1/(d-1)}) (where Õ hides polylogarithmic factors of 1/ε), and also shows the existence of point sets in ℝ^d for which any (1+ε)-spanner must have lightness Ω(ε^{-d}). Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points.
Our first result is a construction of Steiner spanners in ℝ² with lightness O(ε^{-1} log Δ), where Δ is the spread of the point set. In the regime of Δ ≪ 2^(1/ε), this provides an improvement over the lightness bound of [Hung Le and Shay Solomon, 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications ε often controls the precision, and it sometimes needs to be much smaller than O(1/log n). Moreover, for spread polynomially bounded in 1/ε, this upper bound provides a quadratic improvement over the non-Steiner bound of [Hung Le and Shay Solomon, 2019], We then demonstrate that such a light spanner can be constructed in O_ε(n) time for polynomially bounded spread, where O_ε hides a factor of poly(1/(ε)). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(ε^{-(d+1)/2} + ε^{-2} log Δ) for any 3 ≤ d = O(1) and any ε = Ω(n^{-1/(d-1)}).

Hung Le and Shay Solomon. Light Euclidean Spanners with Steiner Points. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 67:1-67:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{le_et_al:LIPIcs.ESA.2020.67, author = {Le, Hung and Solomon, Shay}, title = {{Light Euclidean Spanners with Steiner Points}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {67:1--67:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.67}, URN = {urn:nbn:de:0030-drops-129331}, doi = {10.4230/LIPIcs.ESA.2020.67}, annote = {Keywords: Euclidean spanners, Steiner spanners, light spanners} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Maximal independent set (MIS), maximal matching (MM), and (Delta+1)-(vertex) coloring in graphs of maximum degree Delta are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for (Delta+1)-coloring that runs in O~(n sqrt{n}) time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM.
The neighborhood independence number of a graph G, denoted by beta(G), is the size of the largest independent set in the neighborhood of any vertex. We identify beta(G) as the "right" parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in O(n beta(G)) and O(n log{n} * beta(G)) time respectively on any n-vertex graph G. We complement this positive result by observing that a simple extension of the lower bound of Assadi et al. implies that Omega(n beta(G)) time is also necessary for any algorithm to either problem for all values of beta(G) from 1 to Theta(n). We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires Omega(n^2) time even for beta(G) = 2.
Graphs with bounded neighborhood independence, already for constant beta = beta(G), constitute a rich family of possibly dense graphs, including line graphs, proper interval graphs, unit-disk graphs, claw-free graphs, and graphs of bounded growth. Our results suggest that even though MIS and MM do not admit sublinear-time algorithms in general graphs, one can still solve both problems in sublinear time for a wide range of beta(G) << n.
Finally, by observing that the lower bound of Omega(n sqrt{n}) time for (Delta+1)-coloring due to Assadi et al. applies to graphs of (small) constant neighborhood independence, we unveil an intriguing separation between the time complexity of MIS and MM, and that of (Delta+1)-coloring: while the time complexity of MIS and MM is strictly higher than that of (Delta+1) coloring in general graphs, the exact opposite relation holds for graphs with small neighborhood independence.

Sepehr Assadi and Shay Solomon. When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear Time. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{assadi_et_al:LIPIcs.ICALP.2019.17, author = {Assadi, Sepehr and Solomon, Shay}, title = {{When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear Time}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {17:1--17:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.17}, URN = {urn:nbn:de:0030-drops-105931}, doi = {10.4230/LIPIcs.ICALP.2019.17}, annote = {Keywords: Maximal Independent Set, Maximal Matching, Sublinear-Time Algorithms, Bounded Neighborhood Independence} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

This paper studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n^{1-epsilon} for any epsilon > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring, or alternatively, study restricted families of graphs.
Towards understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for C-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS'17, Barba et al. devised two complementary algorithms: For any beta > 0, the first (respectively, second) maintains an O(C beta n^{1/beta}) (resp., O(C beta))-coloring while recoloring O(beta) (resp., O(beta n^{1/beta})) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for beta = O(1): Any algorithm that maintains a c-coloring of an n-vertex dynamic forest must recolor Omega(n^{2/(c(c-1))}) vertices per update, for any constant c >= 2. Our contribution is two-fold:
- We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: For any beta > 0, we get a O~(C/(beta)log^2 n)-coloring with O(beta) recolorings per update, where the O~ notation supresses polyloglog(n) factors. In particular, for beta = O(1) we get constant recolorings with polylog(n) colors; not only is this an exponential improvement over the previous bound, but it also unveils a rather surprising phenomenon: The trade-off between the number of colors and recolorings is highly non-symmetric.
- For uniformly sparse graphs, we use low out-degree orientations to strengthen the above result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest.

Shay Solomon and Nicole Wein. Improved Dynamic Graph Coloring. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{solomon_et_al:LIPIcs.ESA.2018.72, author = {Solomon, Shay and Wein, Nicole}, title = {{Improved Dynamic Graph Coloring}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {72:1--72:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.72}, URN = {urn:nbn:de:0030-drops-95357}, doi = {10.4230/LIPIcs.ESA.2018.72}, annote = {Keywords: coloring, dynamic graph algorithms, graph arboricity, edge orientations} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The state-of-the-art algorithm for maintaining an approximate maximum matching in fully dynamic graphs has a polynomial worst-case update time, even for poor approximation guarantees. Bhattacharya, Henzinger and Nanongkai showed how to maintain a constant approximation to the minimum vertex cover, and thus also a constant-factor estimate of the maximum matching size, with polylogarithmic worst-case update time. Later (in SODA'17 Proc.) they improved the approximation to 2+epsilon. Nevertheless, the fundamental problem of maintaining an approximate matching with sub-polynomial worst-case time bounds remained open.
We present a randomized algorithm for maintaining an almost-maximal matching in fully dynamic graphs with polylogarithmic worst-case update time. Such a matching provides (2+epsilon)-approximations for both maximum matching and minimum vertex cover, for any epsilon > 0. The worst-case update time of our algorithm, O(poly(log n,epsilon^{-1})), holds deterministically, while the almost-maximality guarantee holds with high probability. Our result was done independently of the (2+epsilon)-approximation result of Bhattacharya et al., thus settling the aforementioned problem on dynamic matchings and providing essentially the best possible approximation guarantee for dynamic vertex cover (assuming the unique games conjecture).
To prove this result, we exploit a connection between the standard oblivious adversarial model, which can be viewed as inherently "online", and an "offline" model where some (limited) information on the future can be revealed efficiently upon demand. Our randomized algorithm is derived from a deterministic algorithm in this offline model. This approach gives an elegant way to analyze randomized dynamic algorithms, and is of independent interest.

Moses Charikar and Shay Solomon. Fully Dynamic Almost-Maximal Matching: Breaking the Polynomial Worst-Case Time Barrier. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{charikar_et_al:LIPIcs.ICALP.2018.33, author = {Charikar, Moses and Solomon, Shay}, title = {{Fully Dynamic Almost-Maximal Matching: Breaking the Polynomial Worst-Case Time Barrier}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.33}, URN = {urn:nbn:de:0030-drops-90370}, doi = {10.4230/LIPIcs.ICALP.2018.33}, annote = {Keywords: dynamic graph algorithms, maximum matching, worst-case bounds} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We consider the problem of maintaining a maximal independent set (MIS) in a dynamic graph subject to edge insertions and deletions. Recently, Assadi, Onak, Schieber and Solomon (STOC 2018) showed that an MIS can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this paper we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity alpha, the amortized update time of our algorithm is O(alpha^2 * log^2 n), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs as well as some classes of "real world" graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m^{3/8 - epsilon}, for any constant epsilon > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m^{1/2}.

Krzysztof Onak, Baruch Schieber, Shay Solomon, and Nicole Wein. Fully Dynamic MIS in Uniformly Sparse Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 92:1-92:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{onak_et_al:LIPIcs.ICALP.2018.92, author = {Onak, Krzysztof and Schieber, Baruch and Solomon, Shay and Wein, Nicole}, title = {{Fully Dynamic MIS in Uniformly Sparse Graphs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {92:1--92:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.92}, URN = {urn:nbn:de:0030-drops-90968}, doi = {10.4230/LIPIcs.ICALP.2018.92}, annote = {Keywords: dynamic graph algorithms, independent set, sparse graphs, graph arboricity} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

In graph sparsification, the goal has almost always been of global nature: compress a graph into a smaller subgraph (sparsifier) that maintains certain features of the original graph.
Algorithms can then run on the sparsifier, which in many cases leads to improvements in the overall runtime and memory.
This paper studies sparsifiers that have bounded (maximum) degree, and are thus locally sparse, aiming to improve local measures of runtime and memory. To improve those local measures, it is important to be able to compute such sparsifiers locally.
We initiate the study of local algorithms for bounded degree sparsifiers in unweighted sparse graphs, focusing on the problems of vertex cover, matching, and independent set. Let \eps > 0 be a slack parameter and \alpha \ge 1 be a density parameter.
We devise local algorithms for computing:
1. A (1+\eps)-vertex cover sparsifier of degree O(\alpha / \eps), for any graph of arboricity \alpha.\footnote{In a graph of arboricity \alpha the average degree of any induced subgraph is at most 2\alpha.}
2. A (1+\eps)-maximum matching sparsifier and also a (1+\eps)-maximal matching sparsifier of degree O(\alpha / \eps, for any graph of arboricity \alpha.
3. A (1+\eps)-independent set sparsifier of degree O(\alpha^2 / \eps), for any graph of average degree \alpha.
Our algorithms require only a single communication round in the standard message passing model of distributed computing,
and moreover, they can be simulated locally in a trivial way.
As an immediate application we can extend results from distributed computing and local computation algorithms that apply to graphs of degree bounded by d to graphs of arboricity O(d / \eps) or average degree O(d^2 / \eps), at the expense of increasing the approximation guarantee by a factor of (1+\eps). In particular, we can extend the plethora of recent local computation algorithms for approximate maximum and maximal matching from bounded degree graphs to bounded arboricity graphs with a negligible loss in the approximation guarantee.
The inherently local behavior of our algorithms can be used to amplify the approximation guarantee of any sparsifier in time roughly linear in its size,
which has immediate applications in the area of dynamic graph algorithms. In particular, the state-of-the-art algorithm for
maintaining (2-\eps)-vertex cover (VC) is at least linear in the graph size, even in dynamic forests. We provide a reduction from the dynamic to the static case, showing that if a t-VC can be computed from scratch in time T(n) in any (sub)family of graphs with arboricity bounded by \alpha, for an arbitrary t \ge 1,
then a (t+\eps)-VC can be maintained with update time \frac{T(n)}{O((n / \alpha) \cdot \eps^2)}, for any \eps > 0. For planar graphs this yields an algorithm for maintaining a (1+\eps)-VC with constant update time for any constant \eps > 0.

Shay Solomon. Local Algorithms for Bounded Degree Sparsifiers in Sparse Graphs. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 52:1-52:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{solomon:LIPIcs.ITCS.2018.52, author = {Solomon, Shay}, title = {{Local Algorithms for Bounded Degree Sparsifiers in Sparse Graphs}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {52:1--52:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.52}, URN = {urn:nbn:de:0030-drops-83647}, doi = {10.4230/LIPIcs.ITCS.2018.52}, annote = {Keywords: arboricity, bounded degree, local algorithm, sparsifier} }

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