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

In an instance of the network design problem, we are given a graph G=(V,E), an edge-cost function c:E -> R^{>= 0}, and a connectivity criterion. The goal is to find a minimum-cost subgraph H of G that meets the connectivity requirements. An important family of this class is the survivable network design problem (SNDP): given non-negative integers r_{u v} for each pair u,v in V, the solution subgraph H should contain r_{u v} edge-disjoint paths for each pair u and v.
While this problem is known to admit good approximation algorithms in the offline case, the problem is much harder in the online setting. Gupta, Krishnaswamy, and Ravi [Gupta et al., 2012] (STOC'09) are the first to consider the online survivable network design problem. They demonstrate an algorithm with competitive ratio of O(k log^3 n), where k=max_{u,v} r_{u v}. Note that the competitive ratio of the algorithm by Gupta et al. grows linearly in k. Since then, an important open problem in the online community [Naor et al., 2011; Gupta et al., 2012] is whether the linear dependence on k can be reduced to a logarithmic dependency.
Consider an online greedy algorithm that connects every demand by adding a minimum cost set of edges to H. Surprisingly, we show that this greedy algorithm significantly improves the competitive ratio when a congestion of 2 is allowed on the edges or when the model is stochastic. While our algorithm is fairly simple, our analysis requires a deep understanding of k-connected graphs. In particular, we prove that the greedy algorithm is O(log^2 n log k)-competitive if one satisfies every demand between u and v by r_{uv}/2 edge-disjoint paths. The spirit of our result is similar to the work of Chuzhoy and Li [Chuzhoy and Li, 2012] (FOCS'12), in which the authors give a polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2.
Moreover, we study the greedy algorithm in the online stochastic setting. We consider the i.i.d. model, where each online demand is drawn from a single probability distribution, the unknown i.i.d. model, where every demand is drawn from a single but unknown probability distribution, and the prophet model in which online demands are drawn from (possibly) different probability distributions. Through a different analysis, we prove that a similar greedy algorithm is constant competitive for the i.i.d. and the prophet models. Also, the greedy algorithm is O(log n)-competitive for the unknown i.i.d. model, which is almost tight due to the lower bound of [Garg et al., 2008] for single connectivity.

Sina Dehghani, Soheil Ehsani, MohammadTaghi Hajiaghayi, Vahid Liaghat, and Saeed Seddighin. Greedy Algorithms for Online Survivable Network Design. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 152:1-152:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dehghani_et_al:LIPIcs.ICALP.2018.152, author = {Dehghani, Sina and Ehsani, Soheil and Hajiaghayi, MohammadTaghi and Liaghat, Vahid and Seddighin, Saeed}, title = {{Greedy Algorithms for Online Survivable Network Design}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {152:1--152:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.152}, URN = {urn:nbn:de:0030-drops-91569}, doi = {10.4230/LIPIcs.ICALP.2018.152}, annote = {Keywords: survivable network design, online, greedy} }

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

In this paper we study a stochastic variant of the celebrated $k$-server problem. In the k-server problem, we are required to minimize the total movement of k servers that are serving an online sequence of $t$ requests in a metric. In the stochastic setting we are given t independent distributions <P_1, P_2, ..., P_t> in advance, and at every time step i a request is drawn from P_i.
Designing the optimal online algorithm in such setting is NP-hard, therefore the emphasis of our work is on designing an approximately optimal online algorithm. We first show a structural characterization for a certain class of non-adaptive online algorithms. We prove that in general metrics, the best of such algorithms has a cost of no worse than three times that of the optimal online algorithm. Next, we present an integer program that finds the optimal algorithm of this class for any arbitrary metric. Finally by rounding the solution of the linear relaxation of this program, we present an online algorithm for the stochastic k-server problem with an approximation factor of $3$ in the line and circle metrics and factor of O(log n) in general metrics. In this way, we achieve an approximation factor that is independent of k, the number of servers.
Moreover, we define the Uber problem, motivated by extraordinary growth of online network transportation services. In the Uber problem, each demand consists of two points -a source and a destination- in the metric. Serving a demand is to move a server to its source and then to its destination. The objective is again minimizing the total movement of the k given servers. It is not hard to show that given an alpha-approximation algorithm for the k-server problem, we can obtain a max{3,alpha}-approximation algorithm for the Uber problem. Motivated by the fact that demands are usually highly correlated with the time (e.g. what day of the week or what time of the day the demand is arrived), we study the stochastic Uber problem. Using our results for stochastic k-server we can obtain a 3-approximation algorithm for the stochastic Uber problem in line and circle metrics, and a O(log n)-approximation algorithm for a general metric of size n.
Furthermore, we extend our results to the correlated setting where the probability of a request arriving at a certain point depends not only on the time step but also on the previously arrived requests.

Sina Dehghani, Soheil Ehsani, MohammadTaghi Hajiaghayi, Vahid Liaghat, and Saeed Seddighin. Stochastic k-Server: How Should Uber Work?. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 126:1-126:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dehghani_et_al:LIPIcs.ICALP.2017.126, author = {Dehghani, Sina and Ehsani, Soheil and Hajiaghayi, MohammadTaghi and Liaghat, Vahid and Seddighin, Saeed}, title = {{Stochastic k-Server: How Should Uber Work?}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {126:1--126:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.126}, URN = {urn:nbn:de:0030-drops-74806}, doi = {10.4230/LIPIcs.ICALP.2017.126}, annote = {Keywords: k-server, stochastic, competitive ratio, online algorithm, Uber} }

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

We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest (EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF, we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance, we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal, we need to augment our solution subgraph to connect the new terminal to the root. The goal is to minimize the total weight of the solution while respecting the degree bounds on the vertices. In the offline setting, edge-weighted degree-bounded Steiner tree (EW-DB-ST) and its many variations have been extensively studied since early eighties. Unfortunately, the recent advancements in the online network design problems are inherently difficult to adapt for degree-bounded problems. In particular, it is not known whether the fractional solution obtained by standard primal-dual techniques for mixed packing/covering LPs can be rounded online. In contrast, in this paper we obtain our result by using structural properties of the optimal solution, and reducing the EW-DB-SF problem to an exponential-size mixed packing/covering integer program in which every variable appears only once in covering constraints. We then design a generic integral algorithm for solving this restricted family of IPs.
As mentioned above, we demonstrate a new technique for solving mixed packing/covering integer programs. Define the covering frequency k of a program as the maximum number of covering constraints in which a variable can participate. Let m denote the number of packing constraints. We design an online deterministic integral algorithm with competitive ratio of O(k*log(m)) for the mixed packing/covering integer programs. We prove the tightness of our result by providing a matching lower bound for any randomized algorithm. We note that our solution solely depends on m and k. Indeed, there can be exponentially many variables. Furthermore, our algorithm directly provides an integral solution, even if the integrality gap of the program is unbounded. We believe this technique can be used as an interesting alternative for the standard primal-dual techniques in solving online problems.

Sina Dehghani, Soheil Ehsani, Mohammad Taghi Hajiaghayi, Vahid Liaghat, Harald Räcke, and Saeed Seddighin. Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 42:1-42:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dehghani_et_al:LIPIcs.ICALP.2016.42, author = {Dehghani, Sina and Ehsani, Soheil and Hajiaghayi, Mohammad Taghi and Liaghat, Vahid and R\"{a}cke, Harald and Seddighin, Saeed}, title = {{Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {42:1--42:14}, 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.2016.42}, URN = {urn:nbn:de:0030-drops-63221}, doi = {10.4230/LIPIcs.ICALP.2016.42}, annote = {Keywords: Online, Steiner Tree, Approximation, Competitive ratio} }

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