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

It is natural to generalize the online k-Server problem by allowing each request to specify not only a point p, but also a subset S of servers that may serve it. To initiate a systematic study of this generalization, we focus on uniform and star metrics. For uniform metrics, the problem is equivalent to a generalization of Paging in which each request specifies not only a page p, but also a subset S of cache slots, and is satisfied by having a copy of p in some slot in S. We call this problem Slot-Heterogenous Paging.
In realistic settings only certain subsets of cache slots or servers would appear in requests. Therefore we parameterize the problem by specifying a family 𝒮 ⊆ 2^[k] of requestable slot sets, and we establish bounds on the competitive ratio as a function of the cache size k and family 𝒮. If all request sets are allowed (𝒮 = 2^[k]), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard Paging (𝒮 = {[k]}). As a function of |𝒮| and k, the optimal deterministic ratio is polynomial: at most O(k²|𝒮|) and at least Ω(√{|𝒮|}). For any laminar family {𝒮} of height h, the optimal ratios are O(hk) (deterministic) and O(h²log k) (randomized). The special case that we call All-or-One Paging extends standard Paging by allowing each request to specify a specific slot to put the requested page in. For All-or-One Paging the optimal competitive ratios are Θ(k) (deterministic) and Θ(log k) (randomized), while the offline problem is NP-hard. We extend the deterministic upper bound to the weighted variant of All-or-One Paging (a generalization of standard Weighted Paging), showing that it is also Θ(k).
Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set P of pages, and is satisfied by fetching any page from P into the cache. The optimal ratios for the latter problem (with laminar family of height h) are at most hk (deterministic) and hH_k (randomized).

Marek Chrobak, Samuel Haney, Mehraneh Liaee, Debmalya Panigrahi, Rajmohan Rajaraman, Ravi Sundaram, and Neal E. Young. Online Paging with Heterogeneous Cache Slots. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chrobak_et_al:LIPIcs.STACS.2023.23, author = {Chrobak, Marek and Haney, Samuel and Liaee, Mehraneh and Panigrahi, Debmalya and Rajaraman, Rajmohan and Sundaram, Ravi and Young, Neal E.}, title = {{Online Paging with Heterogeneous Cache Slots}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {23:1--23:24}, 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.2023.23}, URN = {urn:nbn:de:0030-drops-176759}, doi = {10.4230/LIPIcs.STACS.2023.23}, annote = {Keywords: Caching and paging algorithms, k-server, weighted paging, laminar family} }

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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)

We initiate the algorithmic study of retracting a graph into a cycle in the graph, which seeks a mapping of the graph vertices to the cycle vertices so as to minimize the maximum stretch of any edge, subject to the constraint that the restriction of the mapping to the cycle is the identity map. This problem has its roots in the rich theory of retraction of topological spaces, and has strong ties to well-studied metric embedding problems such as minimum bandwidth and 0-extension. Our first result is an O(min{k, sqrt{n}})-approximation for retracting any graph on n nodes to a cycle with k nodes. We also show a surprising connection to Sperner’s Lemma that rules out the possibility of improving this result using certain natural convex relaxations of the problem. Nevertheless, if the problem is restricted to planar graphs, we show that we can overcome these integrality gaps by giving an optimal combinatorial algorithm, which is the technical centerpiece of the paper. Building on our planar graph algorithm, we also obtain a constant-factor approximation algorithm for retraction of points in the Euclidean plane to a uniform cycle.

Samuel Haney, Mehraneh Liaee, Bruce M. Maggs, Debmalya Panigrahi, Rajmohan Rajaraman, and Ravi Sundaram. Retracting Graphs to Cycles. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{haney_et_al:LIPIcs.ICALP.2019.70, author = {Haney, Samuel and Liaee, Mehraneh and Maggs, Bruce M. and Panigrahi, Debmalya and Rajaraman, Rajmohan and Sundaram, Ravi}, title = {{Retracting Graphs to Cycles}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {70:1--70:15}, 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.2019.70}, URN = {urn:nbn:de:0030-drops-106462}, doi = {10.4230/LIPIcs.ICALP.2019.70}, annote = {Keywords: Graph algorithms, Graph embedding, Planar graphs, Approximation algorithms} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Motivated by denial-of-service network attacks, we introduce the symmetric interdiction model, where both the interdictor and the optimizer are subject to the same constraints of the underlying optimization problem. We give a general framework that relates optimization to symmetric interdiction for a broad class of optimization problems. We then study the symmetric matching interdiction problem - with applications in traffic engineering - in more detail. This problem can be simply stated as follows: find a matching whose removal minimizes the size of the maximum matching in the remaining graph. We show that this problem is APX-hard, and obtain a 3/2-approximation algorithm that improves on the approximation guarantee provided by the general framework.

Samuel Haney, Bruce Maggs, Biswaroop Maiti, Debmalya Panigrahi, Rajmohan Rajaraman, and Ravi Sundaram. Symmetric Interdiction for Matching Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{haney_et_al:LIPIcs.APPROX-RANDOM.2017.9, author = {Haney, Samuel and Maggs, Bruce and Maiti, Biswaroop and Panigrahi, Debmalya and Rajaraman, Rajmohan and Sundaram, Ravi}, title = {{Symmetric Interdiction for Matching Problems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {9:1--9:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.9}, URN = {urn:nbn:de:0030-drops-75587}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.9}, annote = {Keywords: Approximation algorithms, matching, interdiction Digital Object} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

We study the problem of computing a minimum time schedule to spread rumors in a given graph under several models: In the radio model, all neighbors of a transmitting node listen to the messages and are able to record it only when no other neighbor is transmitting; In the wireless model (also called the edge-star model), each transmitter is at a different frequency to which any neighbor can tune to, but only one neighboring transmission can be accessed in this way; In the telephone model, the set of transmitter-receiver pairs form a matching in the graph. The rumor spreading problems assume a message at one or several nodes of the graph that must reach a target node or set of nodes. The transmission proceeds in synchronous rounds under the rules of the corresponding model. The goal is to compute a schedule that completes in the minimum number of rounds.
We present a comprehensive study of approximation algorithms for these problems, and show several reductions from the harder to the easier models for special demands. We show a new hardness of approximation of Omega(n^1/2 - epsilon) for the minimum radio gossip time by a connection to maximum induced matchings. We give the first sublinear approximation algorithms for the most general case of the problem under the wireless model; we also consider various special cases such as instances with symmetric demands and give better approximation algorithms. Our work exposes the relationships across the models and opens up several new avenues for further study.

Jennifer Iglesias, Rajmohan Rajaraman, R. Ravi, and Ravi Sundaram. Rumors Across Radio, Wireless, Telephone. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 517-528, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{iglesias_et_al:LIPIcs.FSTTCS.2015.517, author = {Iglesias, Jennifer and Rajaraman, Rajmohan and Ravi, R. and Sundaram, Ravi}, title = {{Rumors Across Radio, Wireless, Telephone}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {517--528}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.517}, URN = {urn:nbn:de:0030-drops-56383}, doi = {10.4230/LIPIcs.FSTTCS.2015.517}, annote = {Keywords: Broadcast, Gossip, Approximation algorithms, Graph algorithms, Hardness of Approximation} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

From the publish-subscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and IoT (Internet of Things), new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal is a source or sink of information and builds a physical network in the form of a tree or an overlay network in the form of a star rooted at itself. Every pair of pub-sub terminals that need to be coordinated (e.g. the source and sink of an important piece of control information) define an edge in a bipartite demand graph; the solution must ensure that the corresponding networks rooted at the endpoints of each demand edge overlap at some node. This simple overlap constraint, and the requirement that each network is a tree or a star, leads to a variety of new questions on the design of overlapping networks.
In this paper, for the general demand case of the problem, we show that a natural LP formulation has a non-constant integrality gap; on the positive side, we present a logarithmic approximation for the general demand case. When the demand graph is complete, however, we design approximation algorithms with small constant performance ratios, irrespective of whether the pub networks and sub networks are required to be trees or stars.

Jennifer Iglesias, Rajmohan Rajaraman, R. Ravi, and Ravi Sundaram. Designing Overlapping Networks for Publish-Subscribe Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 381-395, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{iglesias_et_al:LIPIcs.APPROX-RANDOM.2015.381, author = {Iglesias, Jennifer and Rajaraman, Rajmohan and Ravi, R. and Sundaram, Ravi}, title = {{Designing Overlapping Networks for Publish-Subscribe Systems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {381--395}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.381}, URN = {urn:nbn:de:0030-drops-53133}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.381}, annote = {Keywords: Approximation Algorithms, Steiner Trees, Publish-Subscribe Systems, Integrality Gap, VPN.} }

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