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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Privately counting distinct elements in a stream is a fundamental data analysis problem with many applications in machine learning. In the turnstile model, Jain et al. [NeurIPS2023] initiated the study of this problem parameterized by the maximum flippancy of any element, i.e., the number of times that the count of an element changes from 0 to above 0 or vice versa. They give an item-level (ε,δ)-differentially private algorithm whose additive error is tight with respect to that parameterization. In this work, we show that a very simple algorithm based on the sparse vector technique achieves a tight additive error for item-level (ε,δ)-differential privacy and item-level ε-differential privacy with regards to a different parameterization, namely the sum of all flippancies. Our second result is a bound which shows that for a large class of algorithms, including all existing differentially private algorithms for this problem, the lower bound from item-level differential privacy extends to event-level differential privacy. This partially answers an open question by Jain et al. [NeurIPS2023].

Monika Henzinger, A. R. Sricharan, and Teresa Anna Steiner. Private Counting of Distinct Elements in the Turnstile Model and Extensions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 40:1-40:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{henzinger_et_al:LIPIcs.APPROX/RANDOM.2024.40, author = {Henzinger, Monika and Sricharan, A. R. and Steiner, Teresa Anna}, title = {{Private Counting of Distinct Elements in the Turnstile Model and Extensions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {40:1--40:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.40}, URN = {urn:nbn:de:0030-drops-210335}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.40}, annote = {Keywords: differential privacy, turnstile model, counting distinct elements} }

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

Oblivious routing is a well-studied paradigm that uses static precomputed routing tables for selecting routing paths within a network. Existing oblivious routing schemes with polylogarithmic competitive ratio for general networks are tree-based, in the sense that routing is performed according to a convex combination of trees. However, this restriction to trees leads to a construction that has time quadratic in the size of the network and does not parallelize well.
In this paper we study oblivious routing schemes based on electrical routing. In particular, we show that general networks with n vertices and m edges admit a routing scheme that has competitive ratio O(log² n) and consists of a convex combination of only O(√m) electrical routings. This immediately leads to an improved construction algorithm with time Õ(m^{3/2}) that can also be implemented in parallel with Õ(√m) depth.

Gramoz Goranci, Monika Henzinger, Harald Räcke, Sushant Sachdeva, and A. R. Sricharan. Electrical Flows for Polylogarithmic Competitive Oblivious Routing. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{goranci_et_al:LIPIcs.ITCS.2024.55, author = {Goranci, Gramoz and Henzinger, Monika and R\"{a}cke, Harald and Sachdeva, Sushant and Sricharan, A. R.}, title = {{Electrical Flows for Polylogarithmic Competitive Oblivious Routing}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {55:1--55:22}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.55}, URN = {urn:nbn:de:0030-drops-195830}, doi = {10.4230/LIPIcs.ITCS.2024.55}, annote = {Keywords: oblivious routing, electrical flows} }

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

Algorithms with predictions is a new research direction that leverages machine learned predictions for algorithm design. So far a plethora of recent works have incorporated predictions to improve on worst-case bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike online algorithms, the goal in dynamic data structures is to maintain the solution efficiently with every update.
We investigate three natural models of prediction: (1) δ-accurate predictions where each predicted request matches the true request with probability δ, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are a permutation of the predicted requests. We give general reductions among the prediction models, showing that bounded delay is the strongest prediction model, followed by list-accurate, and δ-accurate.
Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that locally correctable dynamic problems have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, we show that locally reducible dynamic problems have time complexity that degrades gracefully with the quality of bounded delay predictions. We categorize problems with known OMv lower bounds accordingly and give several upper bounds in the delay model that show that our lower bounds are almost tight.
We note that concurrent work by v.d.Brand et al. [SODA '24] and Liu and Srinivas [arXiv:2307.08890] independently study dynamic graph algorithms with predictions, but their work is mostly focused on showing upper bounds.

Monika Henzinger, Barna Saha, Martin P. Seybold, and Christopher Ye. On the Complexity of Algorithms with Predictions for Dynamic Graph Problems. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 62:1-62:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{henzinger_et_al:LIPIcs.ITCS.2024.62, author = {Henzinger, Monika and Saha, Barna and Seybold, Martin P. and Ye, Christopher}, title = {{On the Complexity of Algorithms with Predictions for Dynamic Graph Problems}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {62:1--62:25}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.62}, URN = {urn:nbn:de:0030-drops-195907}, doi = {10.4230/LIPIcs.ITCS.2024.62}, annote = {Keywords: Dynamic Graph Algorithms, Algorithms with Predictions} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We show an (1+ε)-approximation algorithm for maintaining maximum s-t flow under m edge insertions in m^{1/2+o(1)} ε^{-1/2} amortized update time for directed, unweighted graphs. This constitutes the first sublinear dynamic maximum flow algorithm in general sparse graphs with arbitrarily good approximation guarantee.
Furthermore we give an algorithm that maintains an exact maximum s-t flow under m edge insertions in an n-node graph in Õ(n^{5/2}) total update time. For sufficiently dense graphs, this gives to the first exact incremental algorithm with sub-linear amortized update time for maintaining maximum flows.

Gramoz Goranci and Monika Henzinger. Efficient Data Structures for Incremental Exact and Approximate Maximum Flow. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{goranci_et_al:LIPIcs.ICALP.2023.69, author = {Goranci, Gramoz and Henzinger, Monika}, title = {{Efficient Data Structures for Incremental Exact and Approximate Maximum Flow}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {69:1--69:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.69}, URN = {urn:nbn:de:0030-drops-181212}, doi = {10.4230/LIPIcs.ICALP.2023.69}, annote = {Keywords: dynamic graph algorithms, maximum flow, data structures} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Though tight approximation algorithms for general matroid constraints exist in theory, the running times of such algorithms typically scale quadratically, and are not practical for truly large scale settings. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [Alina Ene and Huy L. Nguyen, 2019]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of 1-1/e-ε and both generalize and accelerate the results of Ene and Nguyen [Alina Ene and Huy L. Nguyen, 2019]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondrák [Ashwinkumar Badanidiyuru and Jan Vondrák, 2014].
To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel Freeze operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [Stephen Alstrup et al., 2005] that maintains the maximum weight basis under insertions and deletions of elements in O(log n) time. This data structure needs to support certain subtree query and path update operations that are performed every insertion and deletion that are non-trivial to handle in conjunction. For the transversal matroid the Freeze operation corresponds to requiring the data structure to keep a certain set S of vertices matched, a property that we call S-stability. While there is a large body of work on dynamic matching algorithms, none are S-stable and maintain an approximate maximum weight matching under vertex updates. We give the first such algorithm for bipartite graphs with total running time linear (up to log factors) in the number of edges.

Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng. Faster Submodular Maximization for Several Classes of Matroids. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 74:1-74:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{henzinger_et_al:LIPIcs.ICALP.2023.74, author = {Henzinger, Monika and Liu, Paul and Vondr\'{a}k, Jan and Zheng, Da Wei}, title = {{Faster Submodular Maximization for Several Classes of Matroids}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {74:1--74:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.74}, URN = {urn:nbn:de:0030-drops-181267}, doi = {10.4230/LIPIcs.ICALP.2023.74}, annote = {Keywords: submodular optimization, dynamic data structures, matching algorithms} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Dynamic programming (DP) is one of the fundamental paradigms in algorithm design. However, many DP algorithms have to fill in large DP tables, represented by two-dimensional arrays, which causes at least quadratic running times and space usages. This has led to the development of improved algorithms for special cases when the DPs satisfy additional properties like, e.g., the Monge property or total monotonicity.
In this paper, we consider a new condition which assumes (among some other technical assumptions) that the rows of the DP table are monotone. Under this assumption, we introduce a novel data structure for computing (1+ε)-approximate DP solutions in near-linear time and space in the static setting, and with polylogarithmic update times when the DP entries change dynamically. To the best of our knowledge, our new condition is incomparable to previous conditions and is the first which allows to derive dynamic algorithms based on existing DPs. Instead of using two-dimensional arrays to store the DP tables, we store the rows of the DP tables using monotone piecewise constant functions. This allows us to store length-n DP table rows with entries in [0,W] using only polylog(n,W) bits, and to perform operations, such as (min,+)-convolution or rounding, on these functions in polylogarithmic time.
We further present several applications of our data structure. For bicriteria versions of k-balanced graph partitioning and simultaneous source location, we obtain the first dynamic algorithms with subpolynomial update times, as well as the first static algorithms using only near-linear time and space. Additionally, we obtain the currently fastest algorithm for fully dynamic knapsack.

Monika Henzinger, Stefan Neumann, Harald Räcke, and Stefan Schmid. Dynamic Maintenance of Monotone Dynamic Programs and Applications. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{henzinger_et_al:LIPIcs.STACS.2023.36, author = {Henzinger, Monika and Neumann, Stefan and R\"{a}cke, Harald and Schmid, Stefan}, title = {{Dynamic Maintenance of Monotone Dynamic Programs and Applications}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {36:1--36:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.36}, URN = {urn:nbn:de:0030-drops-176889}, doi = {10.4230/LIPIcs.STACS.2023.36}, annote = {Keywords: Dynamic programming, dynamic algorithms, data structures} }

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

Data dissemination is a fundamental task in distributed computing. This paper studies broadcast problems in various innovative models where the communication network connecting n processes is dynamic (e.g., due to mobility or failures) and controlled by an adversary.
In the first model, the processes transitively communicate their ids in synchronous rounds along a rooted tree given in each round by the adversary whose goal is to maximize the number of rounds until at least one id is known by all processes. Previous research has shown a ⌈(3n-1)/2⌉-2 lower bound and an O(nlog log n) upper bound. We show the first linear upper bound for this problem, namely ⌈(1+√2) n-1⌉ ≈ 2.4n.
We extend these results to the setting where the adversary gives in each round k-disjoint forests and their goal is to maximize the number of rounds until there is a set of k ids such that each process knows of at least one of them. We give a ⌈3(n-k)/2⌉-1 lower bound and a (π²+6)/6 n+1 ≈ 2.6n upper bound for this problem.
Finally, we study the setting where the adversary gives in each round a directed graph with k roots and their goal is to maximize the number of rounds until there exist k ids that are known by all processes. We give a ⌈3(n-3k)/2⌉+2 lower bound and a ⌈(1+√2)n⌉+k-1 ≈ 2.4n+k upper bound for this problem.
For the two latter problems no upper or lower bounds were previously known.

Antoine El-Hayek, Monika Henzinger, and Stefan Schmid. Asymptotically Tight Bounds on the Time Complexity of Broadcast and Its Variants in Dynamic Networks. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 47:1-47:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{elhayek_et_al:LIPIcs.ITCS.2023.47, author = {El-Hayek, Antoine and Henzinger, Monika and Schmid, Stefan}, title = {{Asymptotically Tight Bounds on the Time Complexity of Broadcast and Its Variants in Dynamic Networks}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {47:1--47:21}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.47}, URN = {urn:nbn:de:0030-drops-175502}, doi = {10.4230/LIPIcs.ITCS.2023.47}, annote = {Keywords: broadcast, cover, k-broadcast, dynamic radius, dynamic graphs, oblivious message adversary, time complexity} }

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

Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form 𝐋𝐱 = 𝐛, where 𝐋 is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution 𝐱 of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i,j) is 1/w(i,j). Kelner, Orrechia, Sidford, and Zhu [Kelner et al., 2013] give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling).
In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations.
We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem (OMv), which has been conjectured to be difficult for dynamic algorithms [Henzinger et al., 2015]. The conjecture implies that the data structure does not have an O(n^{1-ε}) time algorithm for any ε > 0, and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration.
To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an Õ(m^{1.5}) time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O(m^{1 + ε}) for any ε > 0. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart.

Monika Henzinger, Billy Jin, Richard Peng, and David P. Williamson. A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{henzinger_et_al:LIPIcs.ITCS.2023.69, author = {Henzinger, Monika and Jin, Billy and Peng, Richard and Williamson, David P.}, title = {{A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {69:1--69:22}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.69}, URN = {urn:nbn:de:0030-drops-175720}, doi = {10.4230/LIPIcs.ITCS.2023.69}, annote = {Keywords: Laplacian solver, electrical flow, data structure} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and (s,t)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^{1/2 - ε}) update time and O(m^{1 -ε}) query time, for any small ε > 0. Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching.

Monika Henzinger, Ami Paz, and A. R. Sricharan. Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2022.65, author = {Henzinger, Monika and Paz, Ami and Sricharan, A. R.}, title = {{Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {65:1--65:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.65}, URN = {urn:nbn:de:0030-drops-170035}, doi = {10.4230/LIPIcs.ESA.2022.65}, annote = {Keywords: Dynamic graph algorithms, Expander graphs, Power-law graphs} }

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Invited Talk

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We give an overview of differentially private dynamic data structure, aka differentially private algorithms under continual release.

Monika Henzinger. Modern Dynamic Data Structures (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{henzinger:LIPIcs.MFCS.2022.2, author = {Henzinger, Monika}, title = {{Modern Dynamic Data Structures}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {2:1--2:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.2}, URN = {urn:nbn:de:0030-drops-168009}, doi = {10.4230/LIPIcs.MFCS.2022.2}, annote = {Keywords: Differential privacy, data structures} }

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**Published in:** LIPIcs, Volume 218, 3rd Symposium on Foundations of Responsible Computing (FORC 2022)

Finding a representative cohort from a broad pool of candidates is a goal that arises in many contexts such as choosing governing committees and consumer panels. While there are many ways to define the degree to which a cohort represents a population, a very appealing solution concept is lexicographic maximality (leximax) which offers a natural (pareto-optimal like) interpretation that the utility of no population can be increased without decreasing the utility of a population that is already worse off. However, finding a leximax solution can be highly dependent on small variations in the utility of certain groups. In this work, we explore new notions of approximate leximax solutions with three distinct motivations: better algorithmic efficiency, exploiting significant utility improvements, and robustness to noise. Among other definitional contributions, we give a new notion of an approximate leximax that satisfies a similarly appealing semantic interpretation and relate it to algorithmically-feasible approximate leximax notions. When group utilities are linear over cohort candidates, we give an efficient polynomial-time algorithm for finding a leximax distribution over cohort candidates in the exact as well as in the approximate setting. Furthermore, we show that finding an integer solution to leximax cohort selection with linear utilities is NP-Hard.

Monika Henzinger, Charlotte Peale, Omer Reingold, and Judy Hanwen Shen. Leximax Approximations and Representative Cohort Selection. In 3rd Symposium on Foundations of Responsible Computing (FORC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 218, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{henzinger_et_al:LIPIcs.FORC.2022.2, author = {Henzinger, Monika and Peale, Charlotte and Reingold, Omer and Shen, Judy Hanwen}, title = {{Leximax Approximations and Representative Cohort Selection}}, booktitle = {3rd Symposium on Foundations of Responsible Computing (FORC 2022)}, pages = {2:1--2:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-226-6}, ISSN = {1868-8969}, year = {2022}, volume = {218}, editor = {Celis, L. Elisa}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2022.2}, URN = {urn:nbn:de:0030-drops-165258}, doi = {10.4230/LIPIcs.FORC.2022.2}, annote = {Keywords: fairness, cohort selection, leximin, maxmin} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

This paper presents a comprehensive study of algorithms for maintaining the number of all connected four-vertex subgraphs in a dynamic graph. Specifically, our algorithms maintain the number of paths of length three in deterministic amortized O(m^{1/2}) update time, and any other connected four-vertex subgraph which is not a clique in deterministic amortized update time O(m^{2/3}). Queries can be answered in constant time. We also study the query times for subgraphs containing an arbitrary edge that is supplied only with the query as well as the case where only subgraphs containing a vertex s that is fixed beforehand are considered. For length-3 paths, paws, 4-cycles, and diamonds our bounds match or are not far from (conditional) lower bounds: Based on the OMv conjecture we show that any dynamic algorithm that detects the existence of paws, diamonds, or 4-cycles or that counts length-3 paths takes update time Ω(m^{1/2-δ}).
Additionally, for 4-cliques and all connected induced subgraphs, we show a lower bound of Ω(m^{1-δ}) for any small constant δ > 0 for the amortized update time, assuming the static combinatorial 4-clique conjecture holds. This shows that the O(m) algorithm by Eppstein et al. [David Eppstein et al., 2012] for these subgraphs cannot be improved by a polynomial factor.

Kathrin Hanauer, Monika Henzinger, and Qi Cheng Hua. Fully Dynamic Four-Vertex Subgraph Counting. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hanauer_et_al:LIPIcs.SAND.2022.18, author = {Hanauer, Kathrin and Henzinger, Monika and Hua, Qi Cheng}, title = {{Fully Dynamic Four-Vertex Subgraph Counting}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {18:1--18:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.18}, URN = {urn:nbn:de:0030-drops-159608}, doi = {10.4230/LIPIcs.SAND.2022.18}, annote = {Keywords: Dynamic Graph Algorithms, Subgraph Counting, Motif Search} }

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Invited Talk

**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. Here, we present a quick reference guide to recent engineering and theory results in the area of fully dynamic graph algorithms.

Kathrin Hanauer, Monika Henzinger, and Christian Schulz. Recent Advances in Fully Dynamic Graph Algorithms (Invited Talk). In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 1:1-1:47, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hanauer_et_al:LIPIcs.SAND.2022.1, author = {Hanauer, Kathrin and Henzinger, Monika and Schulz, Christian}, title = {{Recent Advances in Fully Dynamic Graph Algorithms}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {1:1--1:47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.1}, URN = {urn:nbn:de:0030-drops-159434}, doi = {10.4230/LIPIcs.SAND.2022.1}, annote = {Keywords: fully dynamic graph algorithms, survey} }

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

Differentially private algorithms protect individuals in data analysis scenarios by ensuring that there is only a weak correlation between the existence of the user in the data and the result of the analysis. Dynamic graph algorithms maintain the solution to a problem (e.g., a matching) on an evolving input, i.e., a graph where nodes or edges are inserted or deleted over time. They output the value of the solution after each update operation, i.e., continuously. We study (event-level and user-level) differentially private algorithms for graph problems under continual observation, i.e., differentially private dynamic graph algorithms. We present event-level private algorithms for partially dynamic counting-based problems such as triangle count that improve the additive error by a polynomial factor (in the length T of the update sequence) on the state of the art, resulting in the first algorithms with additive error polylogarithmic in T.
We also give ε-differentially private and partially dynamic algorithms for minimum spanning tree, minimum cut, densest subgraph, and maximum matching. The additive error of our improved MST algorithm is O(W log^{3/2}T / ε), where W is the maximum weight of any edge, which, as we show, is tight up to a (√{log T} / ε)-factor. For the other problems, we present a partially-dynamic algorithm with multiplicative error (1+β) for any constant β > 0 and additive error O(W log(nW) log(T) / (ε β)). Finally, we show that the additive error for a broad class of dynamic graph algorithms with user-level privacy must be linear in the value of the output solution’s range.

Hendrik Fichtenberger, Monika Henzinger, and Lara Ost. Differentially Private Algorithms for Graphs Under Continual Observation. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fichtenberger_et_al:LIPIcs.ESA.2021.42, author = {Fichtenberger, Hendrik and Henzinger, Monika and Ost, Lara}, title = {{Differentially Private Algorithms for Graphs Under Continual Observation}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {42:1--42:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.42}, URN = {urn:nbn:de:0030-drops-146230}, doi = {10.4230/LIPIcs.ESA.2021.42}, annote = {Keywords: differential privacy, continual observation, dynamic graph algorithms} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

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

Graphs and games on graphs are fundamental models for the analysis of reactive systems, in particular, for model-checking and the synthesis of reactive systems. The class of ω-regular languages provides a robust specification formalism for the desired properties of reactive systems. In the classical infinitary formulation of the liveness part of an ω-regular specification, a "good" event must happen eventually without any bound between the good events. A stronger notion of liveness is bounded liveness, which requires that good events happen within d transitions. Given a graph or a game graph with n vertices, m edges, and a bounded liveness objective, the previous best-known algorithmic bounds are as follows: (i) O(dm) for graphs, which in the worst-case is O(n³); and (ii) O(n² d²) for games on graphs. Our main contributions improve these long-standing algorithmic bounds. For graphs we present: (i) a randomized algorithm with one-sided error with running time O(n^{2.5} log n) for the bounded liveness objectives; and (ii) a deterministic linear-time algorithm for the complement of bounded liveness objectives. For games on graphs, we present an O(n² d) time algorithm for the bounded liveness objectives.

Krishnendu Chatterjee, Monika Henzinger, Sagar Sudhir Kale, and Alexander Svozil. Faster Algorithms for Bounded Liveness in Graphs and Game Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 124:1-124:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chatterjee_et_al:LIPIcs.ICALP.2021.124, author = {Chatterjee, Krishnendu and Henzinger, Monika and Kale, Sagar Sudhir and Svozil, Alexander}, title = {{Faster Algorithms for Bounded Liveness in Graphs and Game Graphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {124:1--124:21}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.124}, URN = {urn:nbn:de:0030-drops-141930}, doi = {10.4230/LIPIcs.ICALP.2021.124}, annote = {Keywords: Graphs, Game Graphs, B\"{u}chi} }

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

With input sizes becoming massive, coresets - small yet representative summary of the input - are relevant more than ever. A weighted set C_w that is a subset of the input is an ε-coreset if the cost of any feasible solution S with respect to C_w is within [1±ε] of the cost of S with respect to the original input. We give a very general technique to compute coresets in the fully-dynamic setting where input points can be added or deleted. Given a static (i.e., not dynamic) ε-coreset-construction algorithm that runs in time t(n, ε, λ) and computes a coreset of size s(n, ε, λ), where n is the number of input points and 1-λ is the success probability, we give a fully-dynamic algorithm that computes an ε-coreset with worst-case update time O((log n) ⋅ t(s(n, ε/log n, λ/n), ε/log n, λ/n)) (this bound is stated informally), where the success probability is 1-λ. Our technique is a fully-dynamic analog of the merge-and-reduce technique, which is due to Har-Peled and Mazumdar [Har-Peled and Mazumdar, 2004] and is based on a technique of Bentley and Saxe [Jon Louis Bentley and James B. Saxe, 1980], that applies to the insertion-only setting where points can only be added. Although, our space usage is O(n), our technique works in the presence of an adaptive adversary, and we show that Ω(n) space is required when adversary is adaptive.
As a concrete implication of our technique, using the result of Braverman et al. [{Braverman} et al., 2016], we get fully-dynamic ε-coreset-construction algorithms for k-median and k-means with worst-case update time O(ε^{-2} k² log⁵ n log³ k) and coreset size O(ε^{-2} k log n log² k) ignoring log log n and log(1/ε) factors and assuming that ε = Ω(1/poly(n)) and λ = Ω(1/poly(n)) (which are very weak assumptions made only to make these bounds easy to parse). This results in the first fully-dynamic constant-approximation algorithms for k-median and k-means with update times O(poly(k, log n, ε^{-1})). Specifically, the dependence on k is only quadratic, and the bounds are worst-case. The best previous bound for both problems was amortized O(nlog n) by Cohen-Addad et al. [Cohen-Addad et al., 2019] via randomized O(1)-coresets in O(n) space.
We also show that under the OMv conjecture [Monika Henzinger et al., 2015], a fully-dynamic (4 - δ)-approximation algorithm for k-means must either have an amortized update time of Ω(k^{1-γ}) or amortized query time of Ω(k^{2 - γ}), where γ > 0 is a constant.

Monika Henzinger and Sagar Kale. Fully-Dynamic Coresets. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 57:1-57:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2020.57, author = {Henzinger, Monika and Kale, Sagar}, title = {{Fully-Dynamic Coresets}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {57:1--57:21}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.57}, URN = {urn:nbn:de:0030-drops-129230}, doi = {10.4230/LIPIcs.ESA.2020.57}, annote = {Keywords: Clustering, Coresets, Dynamic Algorithms} }

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

In recent years, significant advances have been made in the design and analysis of fully dynamic maximal matching algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. In this paper, we attempt to bridge the gap between theory and practice that is currently observed for the fully dynamic maximal matching problem. We engineer several algorithms and empirically study those algorithms on an extensive set of dynamic instances.

Monika Henzinger, Shahbaz Khan, Richard Paul, and Christian Schulz. Dynamic Matching Algorithms in Practice. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 58:1-58:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2020.58, author = {Henzinger, Monika and Khan, Shahbaz and Paul, Richard and Schulz, Christian}, title = {{Dynamic Matching Algorithms in Practice}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {58:1--58:20}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.58}, URN = {urn:nbn:de:0030-drops-129243}, doi = {10.4230/LIPIcs.ESA.2020.58}, annote = {Keywords: Matching, Dynamic Matching, Blossom Algorithm} }

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

We present a practically efficient algorithm that finds all global minimum cuts in huge undirected graphs. Our algorithm uses a multitude of kernelization rules to reduce the graph to a small equivalent instance and then finds all minimum cuts using an optimized version of the algorithm of Nagamochi, Nakao and Ibaraki. In shared memory we are able to find all minimum cuts of graphs with up to billions of edges and millions of minimum cuts in a few minutes. We also give a new linear time algorithm to find the most balanced minimum cuts given as input the representation of all minimum cuts.

Monika Henzinger, Alexander Noe, Christian Schulz, and Darren Strash. Finding All Global Minimum Cuts in Practice. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 59:1-59:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2020.59, author = {Henzinger, Monika and Noe, Alexander and Schulz, Christian and Strash, Darren}, title = {{Finding All Global Minimum Cuts in Practice}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {59:1--59:20}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.59}, URN = {urn:nbn:de:0030-drops-129255}, doi = {10.4230/LIPIcs.ESA.2020.59}, annote = {Keywords: Minimum Cut, Graph Algorithm, Algorithm Engineering, Cut Enumeration, Balanced Cut, Global Minimum Cut, Large-scale Graph Analysis} }

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**Published in:** LIPIcs, Volume 160, 18th International Symposium on Experimental Algorithms (SEA 2020)

The fully dynamic transitive closure problem asks to maintain reachability information in a directed graph between arbitrary pairs of vertices, while the graph undergoes a sequence of edge insertions and deletions. The problem has been thoroughly investigated in theory and many specialized algorithms for solving it have been proposed in the last decades. In two large studies [Frigioni ea, 2001; Krommidas and Zaroliagis, 2008], a number of these algorithms have been evaluated experimentally against simple, static algorithms for graph traversal, showing the competitiveness and even superiority of the simple algorithms in practice, except for very dense random graphs or very high ratios of queries. A major drawback of those studies is that only small and mostly randomly generated graphs are considered.
In this paper, we engineer new algorithms to maintain all-pairs reachability information which are simple and space-efficient. Moreover, we perform an extensive experimental evaluation on both generated and real-world instances that are several orders of magnitude larger than those in the previous studies. Our results indicate that our new algorithms outperform all state-of-the-art algorithms on all types of input considerably in practice.

Kathrin Hanauer, Monika Henzinger, and Christian Schulz. Faster Fully Dynamic Transitive Closure in Practice. In 18th International Symposium on Experimental Algorithms (SEA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 160, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hanauer_et_al:LIPIcs.SEA.2020.14, author = {Hanauer, Kathrin and Henzinger, Monika and Schulz, Christian}, title = {{Faster Fully Dynamic Transitive Closure in Practice}}, booktitle = {18th International Symposium on Experimental Algorithms (SEA 2020)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-148-1}, ISSN = {1868-8969}, year = {2020}, volume = {160}, editor = {Faro, Simone and Cantone, Domenico}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2020.14}, URN = {urn:nbn:de:0030-drops-120887}, doi = {10.4230/LIPIcs.SEA.2020.14}, annote = {Keywords: Dynamic Graph Algorithms, Reachability, Transitive Closure} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Independent set is a fundamental problem in combinatorial optimization. While in general graphs the problem is essentially inapproximable, for many important graph classes there are approximation algorithms known in the offline setting. These graph classes include interval graphs and geometric intersection graphs, where vertices correspond to intervals/geometric objects and an edge indicates that the two corresponding objects intersect.
We present dynamic approximation algorithms for independent set of intervals, hypercubes and hyperrectangles in d dimensions. They work in the fully dynamic model where each update inserts or deletes a geometric object. All our algorithms are deterministic and have worst-case update times that are polylogarithmic for constant d and ε>0, assuming that the coordinates of all input objects are in [0, N]^d and each of their edges has length at least 1. We obtain the following results:
- For weighted intervals, we maintain a (1+ε)-approximate solution.
- For d-dimensional hypercubes we maintain a (1+ε)2^d-approximate solution in the unweighted case and a O(2^d)-approximate solution in the weighted case. Also, we show that for maintaining an unweighted (1+ε)-approximate solution one needs polynomial update time for d ≥ 2 if the ETH holds.
- For weighted d-dimensional hyperrectangles we present a dynamic algorithm with approximation ratio (1+ε)log^{d-1}N.

Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{henzinger_et_al:LIPIcs.SoCG.2020.51, author = {Henzinger, Monika and Neumann, Stefan and Wiese, Andreas}, title = {{Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.51}, URN = {urn:nbn:de:0030-drops-122094}, doi = {10.4230/LIPIcs.SoCG.2020.51}, annote = {Keywords: Dynamic algorithms, independent set, approximation algorithms, interval graphs, geometric intersection graphs} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper (Δ+1)-vertex coloring of a graph with maximum degree at most Δ. This improves upon the previous O(log Δ)-time algorithm by Bhattacharya et al. (SODA 2018). Our algorithm uses an approach based on assigning random ranks to vertices and does not need to maintain a hierarchical graph decomposition. We show that our result does not only have optimal running time, but is also optimal in the sense that already deciding whether a Δ-coloring exists in a dynamically changing graph with maximum degree at most Δ takes Ω(log n) time per operation.

Monika Henzinger and Pan Peng. Constant-Time Dynamic (Δ+1)-Coloring. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{henzinger_et_al:LIPIcs.STACS.2020.53, author = {Henzinger, Monika and Peng, Pan}, title = {{Constant-Time Dynamic (\Delta+1)-Coloring}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {53:1--53:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.53}, URN = {urn:nbn:de:0030-drops-119145}, doi = {10.4230/LIPIcs.STACS.2020.53}, annote = {Keywords: Dynamic graph algorithms, Graph coloring, Random sampling} }

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**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

The fundamental model-checking problem, given as input a model and a specification, asks for the algorithmic verification of whether the model satisfies the specification. Two classical models for reactive systems are graphs and Markov decision processes (MDPs). A basic specification formalism in the verification of reactive systems is the strong fairness (aka Streett) objective, where given different types of requests and corresponding grants, the requirement is that for each type, if the request event happens infinitely often, then the corresponding grant event must also happen infinitely often. All omega-regular objectives can be expressed as Streett objectives and hence they are canonical in verification. Consider graphs/MDPs with n vertices, m edges, and a Streett objectives with k pairs, and let b denote the size of the description of the Streett objective for the sets of requests and grants. The current best-known algorithm for the problem requires time O(min(n^2, m sqrt{m log n}) + b log n). In this work we present randomized near-linear time algorithms, with expected running time O~(m + b), where the O~ notation hides poly-log factors. Our randomized algorithms are near-linear in the size of the input, and hence optimal up to poly-log factors.

Krishnendu Chatterjee, Wolfgang Dvořák, Monika Henzinger, and Alexander Svozil. Near-Linear Time Algorithms for Streett Objectives in Graphs and MDPs. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chatterjee_et_al:LIPIcs.CONCUR.2019.7, author = {Chatterjee, Krishnendu and Dvo\v{r}\'{a}k, Wolfgang and Henzinger, Monika and Svozil, Alexander}, title = {{Near-Linear Time Algorithms for Streett Objectives in Graphs and MDPs}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {7:1--7:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.7}, URN = {urn:nbn:de:0030-drops-109093}, doi = {10.4230/LIPIcs.CONCUR.2019.7}, annote = {Keywords: model checking, graph games, Streett games} }

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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)

The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported.
This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include:
- Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP.
- Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal.

Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein. Algorithms and Hardness for Diameter in Dynamic Graphs. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ancona_et_al:LIPIcs.ICALP.2019.13, author = {Ancona, Bertie and Henzinger, Monika and Roditty, Liam and Williams, Virginia Vassilevska and Wein, Nicole}, title = {{Algorithms and Hardness for Diameter in Dynamic Graphs}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {13:1--13:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.13}, URN = {urn:nbn:de:0030-drops-105891}, doi = {10.4230/LIPIcs.ICALP.2019.13}, annote = {Keywords: fine-grained complexity, graph algorithms, dynamic algorithms} }

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

We study the metric facility location problem with client insertions and deletions. This setting differs from the classic dynamic facility location problem, where the set of clients remains the same, but the metric space can change over time. We show a deterministic algorithm that maintains a constant factor approximation to the optimal solution in worst-case time O~(2^{O(kappa^2)}) per client insertion or deletion in metric spaces while answering queries about the cost in O(1) time, where kappa denotes the doubling dimension of the metric. For metric spaces with bounded doubling dimension, the update time is polylogarithmic in the parameters of the problem.

Gramoz Goranci, Monika Henzinger, and Dariusz Leniowski. A Tree Structure For Dynamic Facility Location. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 39:1-39:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2018.39, author = {Goranci, Gramoz and Henzinger, Monika and Leniowski, Dariusz}, title = {{A Tree Structure For Dynamic Facility Location}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {39:1--39:13}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.39}, URN = {urn:nbn:de:0030-drops-95026}, doi = {10.4230/LIPIcs.ESA.2018.39}, annote = {Keywords: facility location, dynamic algorithm, approximation, doubling dimension} }

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

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an n^{c}-separator theorem for some c<1. We give a fully dynamic algorithm that maintains (1+epsilon)-approximations of the all-pairs effective resistances of an n-vertex graph G undergoing edge insertions and deletions with O~(sqrt{n}/epsilon^2) worst-case update time and O~(sqrt{n}/epsilon^2) worst-case query time, if G is guaranteed to be sqrt{n}-separable (i.e., it is taken from a class satisfying a sqrt{n}-separator theorem) and its separator can be computed in O~(n) time. Our algorithm is built upon a dynamic algorithm for maintaining approximate Schur complement that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest.
We complement our result by proving that for any two fixed vertices s and t, no incremental or decremental algorithm can maintain the s-t effective resistance for sqrt{n}-separable graphs with worst-case update time O(n^{1/2-delta}) and query time O(n^{1-delta}) for any delta>0, unless the Online Matrix Vector Multiplication (OMv) conjecture is false.
We further show that for general graphs, no incremental or decremental algorithm can maintain the s-t effective resistance problem with worst-case update time O(n^{1-delta}) and query-time O(n^{2-delta}) for any delta >0, unless the OMv conjecture is false.

Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2018.40, author = {Goranci, Gramoz and Henzinger, Monika and Peng, Pan}, title = {{Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {40:1--40:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.40}, URN = {urn:nbn:de:0030-drops-95036}, doi = {10.4230/LIPIcs.ESA.2018.40}, annote = {Keywords: Dynamic graph algorithms, effective resistance, separable graphs, Schur complement, conditional lower bounds} }

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**Published in:** LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

It is common knowledge that there is no single best strategy for graph clustering, which justifies a plethora of existing approaches. In this paper, we present a general memetic algorithm, VieClus, to tackle the graph clustering problem. This algorithm can be adapted to optimize different objective functions. A key component of our contribution are natural recombine operators that employ ensemble clusterings as well as multi-level techniques. Lastly, we combine these techniques with a scalable communication protocol, producing a system that is able to compute high-quality solutions in a short amount of time. We instantiate our scheme with local search for modularity and show that our algorithm successfully improves or reproduces all entries of the 10th DIMACS implementation challenge under consideration using a small amount of time.

Sonja Biedermann, Monika Henzinger, Christian Schulz, and Bernhard Schuster. Memetic Graph Clustering. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biedermann_et_al:LIPIcs.SEA.2018.3, author = {Biedermann, Sonja and Henzinger, Monika and Schulz, Christian and Schuster, Bernhard}, title = {{Memetic Graph Clustering}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {3:1--3:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.3}, URN = {urn:nbn:de:0030-drops-89389}, doi = {10.4230/LIPIcs.SEA.2018.3}, annote = {Keywords: Graph Clustering, Evolutionary Algorithms} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Graph games provide the foundation for modeling and synthesis of reactive processes. Such games are played over graphs where the vertices are controlled by two adversarial players. We consider graph games where the objective of the first player is the
conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a mean-payoff condition). There are two variants of the problem, namely, the threshold problem where the quantitative goal is to ensure that the mean-payoff value is above a threshold, and the value problem where the quantitative goal is to ensure the optimal mean-payoff value; in both cases ensuring the qualitative parity objective. The previous best-known algorithms for game graphs with n vertices, m edges,
parity objectives with d priorities, and maximal absolute reward value W for mean-payoff objectives, are as follows: O(n^(d+1)·m·W) for the threshold problem, and O(n^(d+2)·m·W) for the value problem.
Our main contributions are faster algorithms, and the running times of our algorithms are as follows: O(n^(d-1)·m·W) for the threshold problem, and O(n^d·m·W·log(n·W)) for the value problem. For mean-payoff parity objectives with two priorities, our algorithms match the best-known bounds of the algorithms for mean-payoff games (without conjunction with parity objectives). Our results are relevant in synthesis of reactive systems with both functional
requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective).

Krishnendu Chatterjee, Monika Henzinger, and Alexander Svozil. Faster Algorithms for Mean-Payoff Parity Games. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2017.39, author = {Chatterjee, Krishnendu and Henzinger, Monika and Svozil, Alexander}, title = {{Faster Algorithms for Mean-Payoff Parity Games}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.39}, URN = {urn:nbn:de:0030-drops-80809}, doi = {10.4230/LIPIcs.MFCS.2017.39}, annote = {Keywords: graph games, mean-payoff parity games} }

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

In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity.
In this paper we prove conditional lower bounds for these and additional problems in a sensitivity setting. For example, we show that under the Boolean Matrix Multiplication (BMM) conjecture combinatorial algorithms cannot compute the (4/3-\varepsilon)-approximate diameter of an undirected unweighted dense graph with truly subcubic preprocessing time and truly subquadratic update/query time. This result is surprising since in the static setting it is not clear whether a reduction from BMM to diameter is possible. We further show under the BMM conjecture that many problems, such as reachability or approximate shortest paths, cannot be solved faster than by recomputation from scratch even after only one or two edge insertions. We extend our reduction from BMM to Diameter to give a reduction from All Pairs Shortest Paths to Diameter under one deletion in weighted graphs. This is intriguing, as in the static setting it is a big open problem whether Diameter is as hard as APSP. We further get a nearly tight lower bound for shortest paths after two edge deletions based on the APSP conjecture. We give more lower bounds under the Strong Exponential Time Hypothesis. Many of our lower bounds also hold for static oracle data structures where no sensitivity is required.
Finally, we give the first algorithm for the (1+\varepsilon)-approximate radius, diameter, and eccentricity problems in directed or undirected unweighted graphs in case of single edges failures. The algorithm has a truly subcubic running time for graphs with a truly subquadratic number of edges; it is tight w.r.t. the conditional lower bounds we obtain.

Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams. Conditional Hardness for Sensitivity Problems. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 26:1-26:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{henzinger_et_al:LIPIcs.ITCS.2017.26, author = {Henzinger, Monika and Lincoln, Andrea and Neumann, Stefan and Vassilevska Williams, Virginia}, title = {{Conditional Hardness for Sensitivity Problems}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {26:1--26:31}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.26}, URN = {urn:nbn:de:0030-drops-81783}, doi = {10.4230/LIPIcs.ITCS.2017.26}, annote = {Keywords: sensitivity, conditional lower bounds, data structures, dynamic graph algorithms} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Graph Sparsification aims at compressing large graphs into smaller ones while (approximately) preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. Given a weighted graph G=(V,E), and a terminal set K with |K|=k, a quality-q vertex cut sparsifier of G is a graph H with K contained in V_H that preserves the value of minimum cuts separating any bipartition of K, up to a factor of q. We show that planar graphs with all the k terminals lying on the same face admit quality-1 vertex cut sparsifier of size O(k^2) that are also planar. Our result extends to vertex flow and distance sparsifiers. It improves the previous best known bound of O(k^2 2^(2k)) for cut and flow sparsifiers by an exponential factor, and matches an Omega(k^2) lower-bound for this class of graphs.
We also study vertex reachability sparsifiers for directed graphs. Given a digraph G=(V,E) and a terminal set K, a vertex reachability sparsifier of G is a digraph H=(V_H,E_H), K contained in V_H that preserves all reachability information among terminal pairs. We introduce the notion of reachability-preserving minors, i.e., we require H to be a minor of G. Among others, for general planar digraphs, we construct reachability-preserving minors of size O(k^2 log^2 k). We complement our upper-bound by showing that there exists an infinite family of acyclic planar digraphs such that any reachability-preserving minor must have Omega(k^2) vertices.

Gramoz Goranci, Monika Henzinger, and Pan Peng. Improved Guarantees for Vertex Sparsification in Planar Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2017.44, author = {Goranci, Gramoz and Henzinger, Monika and Peng, Pan}, title = {{Improved Guarantees for Vertex Sparsification in Planar Graphs}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {44:1--44:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.44}, URN = {urn:nbn:de:0030-drops-78337}, doi = {10.4230/LIPIcs.ESA.2017.44}, annote = {Keywords: Vertex Sparsification, Graph Sparsification, Planar Graphs, Metric Embedding, Reachability} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms.
Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for (1 + epsilon)-approximating the energy of electrical flows in n-vertex planar graphs with tilde{O}(r epsilon^{-2}) worst-case update time and tilde{O}((r + n / sqrt{r}) epsilon^{-2}) worst-case query time, for any r larger than some constant. For r=n^{2/3}, this gives tilde{O}(n^{2/3} epsilon^{-2}) update time and tilde{O}(n^{2/3} epsilon^{-2}) query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification.
We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest.

Gramoz Goranci, Monika Henzinger, and Pan Peng. The Power of Vertex Sparsifiers in Dynamic Graph Algorithms. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2017.45, author = {Goranci, Gramoz and Henzinger, Monika and Peng, Pan}, title = {{The Power of Vertex Sparsifiers in Dynamic Graph Algorithms}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.45}, URN = {urn:nbn:de:0030-drops-78460}, doi = {10.4230/LIPIcs.ESA.2017.45}, annote = {Keywords: Dynamic graph algorithms, electrical flow, minor-free graphs, max flow} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

In this paper, we study the problem of opening centers to cluster a set of clients in a metric space so as to minimize the sum of the costs of the centers and of the cluster radii, in a dynamic environment where clients arrive and depart, and the solution must be updated efficiently while remaining competitive with respect to the current optimal solution. We call this dynamic sum-of-radii clustering problem.
We present a data structure that maintains a solution whose cost is within a constant factor of the cost of an optimal solution in metric spaces with bounded doubling dimension and whose worst-case update time is logarithmic in the parameters of the problem.

Monika Henzinger, Dariusz Leniowski, and Claire Mathieu. Dynamic Clustering to Minimize the Sum of Radii. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 48:1-48:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2017.48, author = {Henzinger, Monika and Leniowski, Dariusz and Mathieu, Claire}, title = {{Dynamic Clustering to Minimize the Sum of Radii}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {48:1--48:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.48}, URN = {urn:nbn:de:0030-drops-78749}, doi = {10.4230/LIPIcs.ESA.2017.48}, annote = {Keywords: dynamic algorithm, clustering, approximation, doubling dimension} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Graph games with omega-regular winning conditions provide a mathematical framework to analyze a wide range of problems in the analysis of reactive systems and programs (such as the synthesis of reactive systems, program repair, and the verification of branching time properties). Parity conditions are canonical forms to specify omega-regular winning conditions. Graph games with parity conditions are equivalent to mu-calculus model checking, and thus a very important algorithmic problem. Symbolic algorithms are of great significance because they provide scalable algorithms for the analysis of large finite-state systems, as well as algorithms for the analysis of infinite-state systems with finite quotient. A set-based symbolic algorithm uses the basic set operations and the one-step predecessor operators.
We consider graph games with n vertices and parity conditions with c priorities (equivalently, a mu-calculus formula with c alternations of least and greatest fixed points). While many explicit algorithms exist for graph games with parity conditions, for set-based symbolic algorithms there are only two algorithms (notice that we use space to refer to the number of sets stored by a symbolic algorithm): (a) the basic algorithm that requires O(n^c) symbolic operations and linear space; and (b) an improved algorithm that requires O(n^{c/2+1}) symbolic operations but also O(n^{c/2+1}) space (i.e., exponential space).
In this work we present two set-based symbolic algorithms for parity games: (a) our first algorithm requires O(n^{c/2+1}) symbolic operations and only requires linear space; and (b) developing on our first algorithm, we present an algorithm that requires O(n^{c/3+1}) symbolic operations and only linear space. We also present the first linear space set-based symbolic algorithm for parity games that requires at most a sub-exponential number of symbolic operations.

Krishnendu Chatterjee, Wolfgang Dvorák, Monika Henzinger, and Veronika Loitzenbauer. Improved Set-Based Symbolic Algorithms for Parity Games. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatterjee_et_al:LIPIcs.CSL.2017.18, author = {Chatterjee, Krishnendu and Dvor\'{a}k, Wolfgang and Henzinger, Monika and Loitzenbauer, Veronika}, title = {{Improved Set-Based Symbolic Algorithms for Parity Games}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {18:1--18:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.18}, URN = {urn:nbn:de:0030-drops-76830}, doi = {10.4230/LIPIcs.CSL.2017.18}, annote = {Keywords: model checking, graph games, parity games, symbolic computation, progress measure} }

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Invited Talk

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

Fundamental algorithmic problems that lie in the core of many application in formal verification and analysis of systems can be described as graph-related algorithmic problems. Nodes in these problems are of one of two (or three) types, giving rise to a game-theoretic viewpoint: Player one nodes are under the control of the algorithm that wants to accomplish a goal, player two nodes are under the control of a worst-case adversary that tries to keep player one to achieve her goal, and random nodes are under the control of a random process that is oblivious to the goal of player one. A graph containing only player one and random nodes is called a Markov Decision Process, a graph containing only player one and player two nodes is called a game graph. A variety of goals on these graphs are of interest, the simplest being whether a fixed set of nodes can be reached. The algorithmic question is then whether there is a strategy for player one to achieve her goal from a given starting node. In this talk we give an overview of a variety of goals that are interesting in computer-aided verification and present upper and (conditional) lower bounds on the time complexity for deciding whether a winning strategy for player one exists.

Monika Henzinger. Efficient Algorithms for Graph-Related Problems in Computer-Aided Verification (Invited Talk). In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{henzinger:LIPIcs.ICALP.2017.2, author = {Henzinger, Monika}, title = {{Efficient Algorithms for Graph-Related Problems in Computer-Aided Verification}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {2:1--2:1}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.2}, URN = {urn:nbn:de:0030-drops-75054}, doi = {10.4230/LIPIcs.ICALP.2017.2}, annote = {Keywords: Computer-aided Verification, Game Theory, Markov Decision Process} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Given a graph where vertices are partitioned into k terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately. The distortion of a compressed graph is the maximum multiplicative blow-up of distances between all pairs of terminals. We study the trade-off between the number of non-terminals and the distortion. This problem generalizes the Steiner Point Removal (SPR) problem, in which all non-terminals must be removed.
We introduce a novel black-box reduction to convert any lower bound on distortion for the SPR problem into a super-linear lower bound on the number of non-terminals, with the same distortion, for our problem. This allows us to show that there exist graphs such that every minor with distortion less than 2 / 2.5 / 3 must have Omega(k^2) / Omega(k^{5/4}) / Omega(k^{6/5}) non-terminals, plus more trade-offs in between. The black-box reduction has an interesting consequence: if the tight lower bound on distortion for the SPR problem is super-constant, then allowing any O(k) non-terminals will not help improving the lower bound to a constant.
We also build on the existing results on spanners, distance oracles and connected 0-extensions to show a number of upper bounds for general graphs, planar graphs, graphs that exclude a fixed minor and bounded treewidth graphs. Among others, we show that any graph admits a minor with O(log k) distortion and O(k^2) non-terminals, and any planar graph admits a minor with
1 + epsilon distortion and ~O((k/epsilon)^2) non-terminals.

Yun Kuen Cheung, Gramoz Goranci, and Monika Henzinger. Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 131:1-131:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cheung_et_al:LIPIcs.ICALP.2016.131, author = {Cheung, Yun Kuen and Goranci, Gramoz and Henzinger, Monika}, title = {{Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {131:1--131:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.131}, URN = {urn:nbn:de:0030-drops-62675}, doi = {10.4230/LIPIcs.ICALP.2016.131}, annote = {Keywords: Distance Approximating Minor, Graph Minor, Graph Compression, Vertex Sparsification, Metric Embedding} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Games on graphs provide the appropriate framework to study several central problems in computer science, such as verification and synthesis of reactive systems. One of the most basic objectives for games on graphs is the liveness (or Büchi) objective that given a target set of vertices requires that some vertex in the target set is visited infinitely often. We study generalized Büchi objectives (i.e., conjunction of liveness objectives), and implications between two generalized Büchi objectives (known as GR(1) objectives), that arise in numerous applications in computer-aided verification. We present improved algorithms and conditional super-linear lower bounds based on widely believed assumptions about the complexity of (A1) combinatorial Boolean matrix multiplication and (A2) CNF-SAT. We consider graph games with n vertices, m edges, and generalized Büchi objectives with k conjunctions. First, we present an algorithm with running time O(k*n^2), improving the previously known O(k*n*m) and O(k^2*n^2) worst-case bounds. Our algorithm is optimal for dense graphs under (A1). Second, we show that the basic algorithm for the problem is optimal for sparse graphs when the target sets have constant size under (A2). Finally, we consider GR(1) objectives, with k_1 conjunctions in the antecedent and k_2 conjunctions in the consequent, and present an O(k_1 k_2 n^{2.5})-time algorithm, improving the previously known O(k_1*k_2*n*m)-time algorithm for m > n^{1.5}.

Krishnendu Chatterjee, Wolfgang Dvorák, Monika Henzinger, and Veronika Loitzenbauer. Conditionally Optimal Algorithms for Generalized Büchi Games. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chatterjee_et_al:LIPIcs.MFCS.2016.25, author = {Chatterjee, Krishnendu and Dvor\'{a}k, Wolfgang and Henzinger, Monika and Loitzenbauer, Veronika}, title = {{Conditionally Optimal Algorithms for Generalized B\"{u}chi Games}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {25:1--25:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.25}, URN = {urn:nbn:de:0030-drops-64403}, doi = {10.4230/LIPIcs.MFCS.2016.25}, annote = {Keywords: generalized B\"{u}chi objective, GR(1) objective, conditional lower bounds, graph games, graph algorithms, computer-aided verification} }

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

We present a deterministic incremental algorithm for exactly maintaining the size of a minimum cut with ~O(1) amortized time per edge insertion and O(1) query time. This result partially answers an open question posed by Thorup [Combinatorica 2007]. It also stays in sharp contrast to a polynomial conditional lower-bound for the fully-dynamic weighted minimum cut problem. Our algorithm is obtained by combining a recent sparsification technique of Kawarabayashi and Thorup [STOC 2015] and an exact incremental algorithm of Henzinger [J. of Algorithm 1997].
We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an O(n log n/epsilon^2) space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a (1+epsilon)-approximation to the minimum cut. The algorithm has ~O(1) amortized update-time and constant query-time.

Gramoz Goranci, Monika Henzinger, and Mikkel Thorup. Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2016.46, author = {Goranci, Gramoz and Henzinger, Monika and Thorup, Mikkel}, title = {{Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {46:1--46:17}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.46}, URN = {urn:nbn:de:0030-drops-63584}, doi = {10.4230/LIPIcs.ESA.2016.46}, annote = {Keywords: Dynamic Graph Algorithms, Minimum Cut, Edge Connectivity} }

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

During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size d; after the update we have to answer queries.
In this paper, we consider the dynamic subgraph connectivity problem with sensitivity d: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to $d$ vertices are changed. Then we get a sequence of connectivity queries in the subgraph of activated vertices.
We present the first fully dynamic algorithm for this problem which has an update and query time only slightly worse than the best decremental algorithm. In addition, we present the first incremental algorithm which is tight with respect to the best known conditional lower bound; moreover, the algorithm is simple and we believe it is implementable and efficient in practice.

Monika Henzinger and Stefan Neumann. Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 48:1-48:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2016.48, author = {Henzinger, Monika and Neumann, Stefan}, title = {{Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {48:1--48:11}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.48}, URN = {urn:nbn:de:0030-drops-63607}, doi = {10.4230/LIPIcs.ESA.2016.48}, annote = {Keywords: connectivity, emergency planning, sensitivity} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Online social networks allow the collection of large amounts of data about the influence between users connected by a friendship-like relationship. When distributing items among agents forming a social network, this information allows us to exploit network externalities that each agent receives from his neighbors that get the same item. In this paper we consider Friends-of-Friends (2-hop) network externalities, i.e., externalities that not only depend on the neighbors that get the same item but also on neighbors of neighbors. For these externalities we study a setting where multiple different items are assigned to unit-demand agents. Specifically, we study the problem of welfare maximization under different types of externality functions. Let n be the number of agents and m be the number of items. Our contributions are the following: (1) We show that welfare maximization is APX-hard; we show that even for step functions with 2-hop (and also with 1-hop) externalities it is NP-hard to approximate social welfare better than (1-1/e). (2) On the positive side we present (i) an O(sqrt n)-approximation algorithm for general concave externality functions,
(ii) an O(\log m)-approximation algorithm for linear externality functions, and (iii) an (1-1/e)\frac{1}{6}-approximation algorithm for 2-hop step function externalities. We also improve the result from [6] for 1-hop step function externalities by giving a (1-1/e)/2-approximation algorithm.

Sayan Bhattacharya, Wolfgang Dvorák, Monika Henzinger, and Martin Starnberger. Welfare Maximization with Friends-of-Friends Network Externalities. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 90-102, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bhattacharya_et_al:LIPIcs.STACS.2015.90, author = {Bhattacharya, Sayan and Dvor\'{a}k, Wolfgang and Henzinger, Monika and Starnberger, Martin}, title = {{Welfare Maximization with Friends-of-Friends Network Externalities}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {90--102}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.90}, URN = {urn:nbn:de:0030-drops-49066}, doi = {10.4230/LIPIcs.STACS.2015.90}, annote = {Keywords: network externalities, welfare maximization, approximation algorithms} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Two-sided matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market where $n$ advertisers compete for the assignment of one of $k$ sponsored search results, also known as ``slots'', for certain keywords they are interested in. Here, as in other markets of that kind, market equilibria correspond to stable matchings. In this paper, we show how to modify Kuhn's Hungarian Method (Kuhn, 1955) so that it finds an optimal stable matching between advertisers and advertising slots in settings with generalized linear utilities, per-bidder-item reserve prices, and per-bidder-item maximum prices. The only algorithm for this problem presented so far (Aggarwal et al., 2009) requires the market to be in ``general position''. We do not make this assumption.

Paul Dütting, Monika Henzinger, and Ingmar Weber. Sponsored Search, Market Equilibria, and the Hungarian Method. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 287-298, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dutting_et_al:LIPIcs.STACS.2010.2463, author = {D\"{u}tting, Paul and Henzinger, Monika and Weber, Ingmar}, title = {{Sponsored Search, Market Equilibria, and the Hungarian Method}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {287--298}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2463}, URN = {urn:nbn:de:0030-drops-24636}, doi = {10.4230/LIPIcs.STACS.2010.2463}, annote = {Keywords: Stable matching, truthful matching mechanism, general position} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

As the World Wide Web is growing rapidly, it is getting increasingly challenging to gather representative information about it. Instead of crawling the web exhaustively one has to resort to other techniques like sampling to determine the properties of the web. A uniform random sample of the web would be useful to determine the percentage of web pages in a specific language, on a topic or in a top level domain. Unfortunately, no approach has been shown to sample the web pages in an unbiased way. Three promising web sampling algorithms are based on random walks. They each have been evaluated individually, but making a comparison on different data sets is not possible. We directly compare these algorithms in this paper. We performed three random walks on the web under the same conditions and analyzed their outcomes in detail. We discuss the strengths and the weaknesses of each algorithm and propose improvements based on experimental results.

Eda Baykan, Monika Henzinger, Stefan F. Keller, Sebastian de Castelberg, and Markus Kinzler. A Comparison of Techniques for Sampling Web Pages. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 13-30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{baykan_et_al:LIPIcs.STACS.2009.1809, author = {Baykan, Eda and Henzinger, Monika and Keller, Stefan F. and de Castelberg, Sebastian and Kinzler, Markus}, title = {{A Comparison of Techniques for Sampling Web Pages}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {13--30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1809}, URN = {urn:nbn:de:0030-drops-18091}, doi = {10.4230/LIPIcs.STACS.2009.1809}, annote = {Keywords: Random walks, Sampling web pages} }

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