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

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence 2-approximate) matching in O(n) worst-case update time in n-node graphs.
We present the first deterministic algorithm which outperforms the folklore algorithm in terms of both approximation ratio and worst-case update time. Specifically, we give a (2-Ω(1))-approximate algorithm with O(m^{3/8}) = O(n^{3/4}) worst-case update time in n-node, m-edge graphs. For sufficiently small constant ε > 0, no deterministic (2+ε)-approximate algorithm with worst-case update time O(n^{0.99}) was known. Our second result is the first deterministic (2+ε)-approximate weighted matching algorithm with O_ε(1)⋅ O(∜{m}) = O_ε(1)⋅ O(√n) worst-case update time. Neither of our results were previously known to be achievable by a randomized algorithm against an adaptive adversary.
Our main technical contributions are threefold: first, we characterize the tight cases for kernels, which are the well-studied matching sparsifiers underlying much of the (2+ε)-approximate dynamic matching literature. This characterization, together with multiple ideas - old and new - underlies our result for breaking the approximation barrier of 2. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the recourse of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural 3/2 factor in the approximation ratio which such approaches naturally incur (reminiscent of the integrality gap of the fractional matching polytope in general graphs).

Mohammad Roghani, Amin Saberi, and David Wajc. Beating the Folklore Algorithm for Dynamic Matching. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 111:1-111:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{roghani_et_al:LIPIcs.ITCS.2022.111, author = {Roghani, Mohammad and Saberi, Amin and Wajc, David}, title = {{Beating the Folklore Algorithm for Dynamic Matching}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {111:1--111:23}, 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.2022.111}, URN = {urn:nbn:de:0030-drops-157077}, doi = {10.4230/LIPIcs.ITCS.2022.111}, annote = {Keywords: dynamic matching, dynamic graph algorithms, sublinear algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of 2 of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree Δ = O(log n), which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or for random arrival orders, an answer for fully general adversarial arrivals remained elusive.
We resolve this thirty-year-old conjecture in the affirmative, presenting a (1.9+o(1))-competitive online edge coloring algorithm for general graphs of degree Δ = ω(log n) under vertex arrivals. At the core of our results, and of possible independent interest, is a new online algorithm which rounds a fractional bipartite matching x online under vertex arrivals, guaranteeing that each edge e is matched with probability (1/2+c)⋅ x_e, for a constant c > 0.027.

Amin Saberi and David Wajc. The Greedy Algorithm Is not Optimal for On-Line Edge Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 109:1-109:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{saberi_et_al:LIPIcs.ICALP.2021.109, author = {Saberi, Amin and Wajc, David}, title = {{The Greedy Algorithm Is not Optimal for On-Line Edge Coloring}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {109:1--109:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.109}, URN = {urn:nbn:de:0030-drops-141786}, doi = {10.4230/LIPIcs.ICALP.2021.109}, annote = {Keywords: Online algorithms, edge coloring, greedy, online matching} }

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

We study the stable marriage problem in two-sided markets with randomly generated preferences. Agents on each side of the market are divided into a constant number of "soft" tiers, which capture agents' qualities. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side.
We compute the expected average rank which agents in each tier have for their partners in the man-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes the results by Pittel [Pittel, 1989], which analyzed markets with uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market.

Itai Ashlagi, Mark Braverman, Amin Saberi, Clayton Thomas, and Geng Zhao. Tiered Random Matching Markets: Rank Is Proportional to Popularity. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ashlagi_et_al:LIPIcs.ITCS.2021.46, author = {Ashlagi, Itai and Braverman, Mark and Saberi, Amin and Thomas, Clayton and Zhao, Geng}, title = {{Tiered Random Matching Markets: Rank Is Proportional to Popularity}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {46:1--46:16}, 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.2021.46}, URN = {urn:nbn:de:0030-drops-135851}, doi = {10.4230/LIPIcs.ITCS.2021.46}, annote = {Keywords: Stable matching, stable marriage problem, tiered random markets, deferred acceptance} }

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

We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tutte, 1948], and sampling can be reduced to counting [Jerrum et al., 1986; Jerrum and Sinclair, 1996]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC.
For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csanky, 1975]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.

Nima Anari, Nathan Hu, Amin Saberi, and Aaron Schild. Sampling Arborescences in Parallel. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 83:1-83:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2021.83, author = {Anari, Nima and Hu, Nathan and Saberi, Amin and Schild, Aaron}, title = {{Sampling Arborescences in Parallel}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {83:1--83:18}, 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.2021.83}, URN = {urn:nbn:de:0030-drops-136225}, doi = {10.4230/LIPIcs.ITCS.2021.83}, annote = {Keywords: parallel algorithms, arborescences, spanning trees, random sampling} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We study the problem of allocating m items to n agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a 1/e approximation factor of the objective, breaking the 1/2e^(1/e) approximation factor of Cole and Gkatzelis.
Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree m-homogeneous stable polynomial on m variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.

Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. Nash Social Welfare, Matrix Permanent, and Stable Polynomials. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2017.36, author = {Anari, Nima and Oveis Gharan, Shayan and Saberi, Amin and Singh, Mohit}, title = {{Nash Social Welfare, Matrix Permanent, and Stable Polynomials}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {36:1--36:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.36}, URN = {urn:nbn:de:0030-drops-81489}, doi = {10.4230/LIPIcs.ITCS.2017.36}, annote = {Keywords: Nash Welfare, Permanent, Matching, Stable Polynomial, Randomized Algorithm, Saddle Point} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Motivated by the importance of energy storage networks in smart grids, we provide an algorithmic study of the online energy storage management problem in a network setting, the first to the best of our knowledge. Given online power supplies, either entirely renewable supplies or those in combination with traditional supplies, we want to route power from the supplies to demands using storage units subject to a decay factor. Our goal is to maximize the total utility of satisfied demands less the total production cost of routed power. We model renewable supplies with the zero production cost function and traditional supplies with convex production cost functions. For two natural storage unit settings, private and public, we design poly-logarithmic competitive algorithms in the network flow model using the dual fitting and online primal dual methods for convex problems. Furthermore, we show strong hardness results for more general settings of the problem. Our techniques may be of independent interest in other routing and storage management problems.

Anthony Kim, Vahid Liaghat, Junjie Qin, and Amin Saberi. Online Energy Storage Management: an Algorithmic Approach. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 12:1-12:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kim_et_al:LIPIcs.APPROX-RANDOM.2016.12, author = {Kim, Anthony and Liaghat, Vahid and Qin, Junjie and Saberi, Amin}, title = {{Online Energy Storage Management: an Algorithmic Approach}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {12:1--12:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.12}, URN = {urn:nbn:de:0030-drops-66355}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.12}, annote = {Keywords: Online Algorithms, Competitive Analysis, Routing, Storage, Approximation Algorithms, Power Control} }

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