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

We study the following basic problem called Bi-Covering. Given a graph G(V, E), find two (not necessarily disjoint) sets A subseteq V and B subseteq V such that A union B = V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize max{|A|, |B|}. This is the most simple case of the Channel Allocation problem [Gandhi et al., Networks, 2006]. A solution that outputs V,emptyset gives ratio at most 2. We show that under the similar Strong Unique Game Conjecture by [Bansal-Khot, FOCS, 2009] there is no 2 - epsilon ratio algorithm for the problem, for any constant epsilon > 0.
Given a bipartite graph, Max-bi-clique is a problem of finding largest k*k complete bipartite sub graph. For Max-bi-clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [Feige, STOC, 2002] and also under the assumption that NP !subseteq intersection_{epsilon > 0} BPTIME(2^{n^{epsilon}}) [Khot, SIAM J. on Comp., 2011]. It was an open problem in [Ambühl et. al., SIAM J. on Comp., 2011] to prove inapproximability of Max-bi-clique assuming weaker conjecture. Our result implies similar hardness result assuming the Strong Unique Games Conjecture.
On the algorithmic side, we also give better than 2 approximation for Bi-Covering on numerous special graph classes. In particular, we get 1.876 approximation for Chordal graphs, exact algorithm for Interval Graphs, 1 + o(1) for Minor Free Graph, 2 - 4*delta/3 for graphs with minimum degree delta*n, 2/(1+delta^2/8) for delta-vertex expander, 8/5 for Split Graphs, 2 - (6/5)*1/d for graphs with minimum constant degree d etc. Our algorithmic results are quite non-trivial. In achieving these results, we use various known structural results about the graphs, combined with the techniques that we develop tailored to getting better than 2 approximation.

Amey Bhangale, Rajiv Gandhi, Mohammad Taghi Hajiaghayi, Rohit Khandekar, and Guy Kortsarz. Bicovering: Covering Edges With Two Small Subsets of Vertices. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bhangale_et_al:LIPIcs.ICALP.2016.6, author = {Bhangale, Amey and Gandhi, Rajiv and Hajiaghayi, Mohammad Taghi and Khandekar, Rohit and Kortsarz, Guy}, title = {{Bicovering: Covering Edges With Two Small Subsets of Vertices}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {6:1--6:12}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.6}, URN = {urn:nbn:de:0030-drops-62728}, doi = {10.4230/LIPIcs.ICALP.2016.6}, annote = {Keywords: Bi-covering, Unique Games, Max Bi-clique} }

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

We study competition in a general framework introduced by Immorlica, Kalai, Lucier, Moitra, Postlewaite, and Tennenholtz and answer their main open question. Immorlica et al. considered classic optimization problems in terms of competition and introduced a general class of games called dueling games. They model this competition as a zero-sum game, where two players are competing for a user’s satisfaction. In their main and most natural game, the ranking duel, a user requests a webpage by submitting a query and players output an ordering over all possible webpages based on the submitted query. The user tends to choose the ordering which displays her requested webpage in a higher rank. The goal of both players is to maximize the probability that her ordering beats that of her opponent and gets the user's attention. Immorlica et al. show this game directs both players to provide suboptimal search results. However, they leave the following as their main open question: "does competition between algorithms improve or degrade expected performance?" (see the introduction for more quotes) In this paper, we resolve this question for the ranking duel and a more general class of dueling games.
More precisely, we study the quality of orderings in a competition between two players. This game is a zero-sum game, and thus any Nash equilibrium of the game can be described by minimax strategies. Let the value of the user for an ordering be a function of the position of her requested item in the corresponding ordering, and the social welfare for an ordering be the expected value of the corresponding ordering for the user. We propose the price of competition which is the ratio of the social welfare for the worst minimax strategy to the social welfare obtained by asocial planner. Finding the price of competition is another approach to obtain structural results of Nash equilibria. We use this criterion for analyzing the quality of orderings in the ranking duel. Although Immorlica et al. show that the competition leads to suboptimal strategies, we prove the quality of minimax results is surprisingly close to that of the optimum solution. In particular, via a novel factor-revealing LP for computing price of anarchy, we prove if the value of the user for an ordering is a linear function of its position, then the price of competition is at least 0.612 and bounded above by 0.833. Moreover we consider the cost minimization version of the problem. We prove, the social cost of the worst minimax strategy is at most 3 times the optimal social cost.
Last but not least, we go beyond linear valuation functions and capture the main challenge for bounding the price of competition for any arbitrary valuation function. We present a principle which states that the lower bound for the price of competition for all 0-1 valuation functions is the same as the lower bound for the price of competition for all possible valuation functions. It is worth mentioning that this principle not only works for the ranking duel but also for all dueling games. This principle says, in any dueling game, the most challenging part of bounding the price of competition is finding a lower bound for 0-1 valuation functions. We leverage this principle to show that the price of competition is at least 0.25 for the generalized ranking duel.

Sina Dehghani, Mohammad Taghi Hajiaghayi, Hamid Mahini, and Saeed Seddighin. Price of Competition and Dueling Games. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dehghani_et_al:LIPIcs.ICALP.2016.21, author = {Dehghani, Sina and Hajiaghayi, Mohammad Taghi and Mahini, Hamid and Seddighin, Saeed}, title = {{Price of Competition and Dueling Games}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {21:1--21:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.21}, URN = {urn:nbn:de:0030-drops-63009}, doi = {10.4230/LIPIcs.ICALP.2016.21}, annote = {Keywords: POC, POA, Dueling games, Nash equilibria, sponsored search} }

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

Miller and Reif’s FOCS'85 [Gary L. Miller and John H. Reif, 1989] classic and fundamental tree contraction algorithm is a broadly applicable technique for the parallel solution of a large number of tree problems. Additionally it is also used as an algorithmic design technique for a large number of parallel graph algorithms. In all previously explored models of computation, however, tree contractions have only been achieved in Ω(log n) rounds of parallel run time. In this work, we not only introduce a generalized tree contraction method but also show it can be computed highly efficiently in O(1/ε³) rounds in the Adaptive Massively Parallel Computing (AMPC) setting, where each machine has O(n^ε) local memory for some 0 < ε < 1. AMPC is a practical extension of Massively Parallel Computing (MPC) which utilizes distributed hash tables [MohammadHossein Bateni et al., 2017; Behnezhad et al., 2019; Raimondas Kiveris et al., 2014]. In general, MPC is an abstract model for MapReduce, Hadoop, Spark, and Flume which are currently widely used across industry and has been studied extensively in the theory community in recent years. Last but not least, we show that our results extend to multiple problems on trees, including but not limited to maximum and maximal matching, maximum and maximal independent set, tree isomorphism testing, and more.

MohammadTaghi Hajiaghayi, Marina Knittel, Hamed Saleh, and Hsin-Hao Su. Adaptive Massively Parallel Constant-Round Tree Contraction. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 83:1-83:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hajiaghayi_et_al:LIPIcs.ITCS.2022.83, author = {Hajiaghayi, MohammadTaghi and Knittel, Marina and Saleh, Hamed and Su, Hsin-Hao}, title = {{Adaptive Massively Parallel Constant-Round Tree Contraction}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {83:1--83:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.83}, URN = {urn:nbn:de:0030-drops-156790}, doi = {10.4230/LIPIcs.ITCS.2022.83}, annote = {Keywords: Adaptive Massively Parallel Computation, Tree Contraction, Matching, Independent Set, Tree Isomorphism} }

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Track A: Algorithms, Complexity and Games

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

The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we first consider these problems in the asymmetric streaming model introduced by Andoni, Krauthgamer and Onak [Andoni et al., 2010] (FOCS'10) and Saks and Seshadhri [Saks and Seshadhri, 2013] (SODA'13). In this model we have random access to one string and streaming access the other one. Our main contribution is a constant factor approximation algorithm for ED with memory Õ(n^δ) for any constant δ > 0. In addition to this, we present an upper bound of Õ _ε(√n) on the memory needed to approximate ED or LCS within a factor 1±ε. All our algorithms are deterministic and run in polynomial time in a single pass.
We further study small-space approximation algorithms for ED, LCS, and longest increasing sequence (LIS) in the non-streaming setting. Here, we design algorithms that achieve 1 ± ε approximation for all three problems, where ε > 0 can be any constant and even slightly sub-constant. Our algorithms only use poly-logarithmic space while maintaining a polynomial running time. This significantly improves previous results in terms of space complexity, where all known results need to use space at least Ω(√n). Our algorithms make novel use of triangle inequality and carefully designed recursions to save space, which can be of independent interest.

Kuan Cheng, Alireza Farhadi, MohammadTaghi Hajiaghayi, Zhengzhong Jin, Xin Li, Aviad Rubinstein, Saeed Seddighin, and Yu Zheng. Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 54:1-54:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cheng_et_al:LIPIcs.ICALP.2021.54, author = {Cheng, Kuan and Farhadi, Alireza and Hajiaghayi, MohammadTaghi and Jin, Zhengzhong and Li, Xin and Rubinstein, Aviad and Seddighin, Saeed and Zheng, Yu}, title = {{Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {54:1--54:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.54}, URN = {urn:nbn:de:0030-drops-141236}, doi = {10.4230/LIPIcs.ICALP.2021.54}, annote = {Keywords: Edit Distance, Longest Common Subsequence, Longest Increasing Subsequence, Space Efficient Algorithm, Approximation Algorithm} }

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Brief Announcement

**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. In this paper, we revisit this problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model:
Massively Parallel Computation. We give a randomized MPC algorithm that w.h.p., returns a (1+o(1))Delta edge coloring in O(1) rounds using O~(n) space per machine and O(m) total space. The space per machine can also be further improved to n^{1-Omega(1)} if Delta = n^{Omega(1)}. This is, to our knowledge, the first constant round algorithm for a natural graph problem in the strongly sublinear regime of MPC. Our algorithm improves a previous result of Harvey et al. [SPAA 2018] which required n^{1+Omega(1)} space to achieve the same result.
Graph Streaming. Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors.
We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors w.h.p., if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors.

Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Marina Knittel, and Hamed Saleh. Brief Announcement: Streaming and Massively Parallel Algorithms for Edge Coloring. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 36:1-36:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{behnezhad_et_al:LIPIcs.DISC.2019.36, author = {Behnezhad, Soheil and Derakhshan, Mahsa and Hajiaghayi, MohammadTaghi and Knittel, Marina and Saleh, Hamed}, title = {{Brief Announcement: Streaming and Massively Parallel Algorithms for Edge Coloring}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {36:1--36:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.36}, URN = {urn:nbn:de:0030-drops-113438}, doi = {10.4230/LIPIcs.DISC.2019.36}, annote = {Keywords: Massively Parallel Computation, Streaming, Edge Coloring} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model:
- Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018].
- Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors.
We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors.

Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Marina Knittel, and Hamed Saleh. Streaming and Massively Parallel Algorithms for Edge Coloring. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{behnezhad_et_al:LIPIcs.ESA.2019.15, author = {Behnezhad, Soheil and Derakhshan, Mahsa and Hajiaghayi, MohammadTaghi and Knittel, Marina and Saleh, Hamed}, title = {{Streaming and Massively Parallel Algorithms for Edge Coloring}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {15:1--15:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.15}, URN = {urn:nbn:de:0030-drops-111361}, doi = {10.4230/LIPIcs.ESA.2019.15}, annote = {Keywords: Massively Parallel Computation, Streaming, Edge Coloring} }

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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-dev.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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Brief Announcement

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

Solving large-scale graph problems is a fundamental task in many real-world applications, and it is an increasingly important problem in data analysis. Despite the large effort in designing scalable graph algorithms, many classic graph problems lack algorithms that require only a sublinear number of machines and space in the input size. Specifically when the input graph is large and sparse, which is indeed the case for many real-world graphs, it becomes impossible to store and access all the vertices in one machine - something that is often taken for granted in designing algorithms for massive graphs. The theoretical model that we consider is the Massively Parallel Communications (MPC) model which is a popular theoretical model of MapReduce-like systems. In this paper, we give an algorithmic framework to adapt a large family of dynamic programs on MPC. We start by introducing two classes of dynamic programming problems, namely "(poly log)-expressible" and "linear-expressible" problems. We show that both classes can be solved efficiently using a sublinear number of machines and a sublinear memory per machine. To achieve this result, we introduce a series of techniques that can be plugged together. To illustrate the generality of our framework, we implement in O(log n) rounds of MPC, the dynamic programming solution of fundamental problems such as minimum bisection, k-spanning tree, maximum independent set, longest path, etc., when the input graph is a tree.

MohammadHossein Bateni, Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, and Vahab Mirrokni. Brief Announcement: MapReduce Algorithms for Massive Trees. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 162:1-162:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bateni_et_al:LIPIcs.ICALP.2018.162, author = {Bateni, MohammadHossein and Behnezhad, Soheil and Derakhshan, Mahsa and Hajiaghayi, MohammadTaghi and Mirrokni, Vahab}, title = {{Brief Announcement: MapReduce Algorithms for Massive Trees}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {162:1--162:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.162}, URN = {urn:nbn:de:0030-drops-91666}, doi = {10.4230/LIPIcs.ICALP.2018.162}, annote = {Keywords: MapReduce, Trees} }

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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-dev.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 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem.
Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs).
(1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014).
(2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as \emph{matching coverage}, in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied.
Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5.

Melika Abolhassani, T.-H. Hubert Chan, Fei Chen, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Mahini Hamid, and Xiaowei Wu. Beating Ratio 0.5 for Weighted Oblivious Matching Problems. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abolhassani_et_al:LIPIcs.ESA.2016.3, author = {Abolhassani, Melika and Chan, T.-H. Hubert and Chen, Fei and Esfandiari, Hossein and Hajiaghayi, MohammadTaghi and Hamid, Mahini and Wu, Xiaowei}, title = {{Beating Ratio 0.5 for Weighted Oblivious Matching Problems}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {3:1--3:18}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.3}, URN = {urn:nbn:de:0030-drops-63443}, doi = {10.4230/LIPIcs.ESA.2016.3}, annote = {Keywords: Weighted matching, oblivious algorithms, Ranking, linear programming} }

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**Published in:** Dagstuhl Reports, Volume 3, Issue 3 (2013)

We provide a report on the Dagstuhl Seminar 13121: "Bidimensional Structures: Algorithms, Combinatorics and Logic" held at Schloss Dagstuhl in Wadern, Germany
between Monday 18 and Friday 22 of March 2013. The report contains the motivation of the seminar, the abstracts of the talks given during the seminar, and the list of open problems.

Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensional Structures: Algorithms, Combinatorics and Logic (Dagstuhl Seminar 13121). In Dagstuhl Reports, Volume 3, Issue 3, pp. 51-74, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{demaine_et_al:DagRep.3.3.51, author = {Demaine, Erik D. and Fomin, Fedor V. and Hajiaghayi, MohammadTaghi and Thilikos, Dimitrios M.}, title = {{Bidimensional Structures: Algorithms, Combinatorics and Logic (Dagstuhl Seminar 13121)}}, pages = {51--74}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {3}, number = {3}, editor = {Demaine, Erik D. and Fomin, Fedor V. and Hajiaghayi, MohammadTaghi and Thilikos, Dimitrios M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.3.3.51}, URN = {urn:nbn:de:0030-drops-40131}, doi = {10.4230/DagRep.3.3.51}, annote = {Keywords: Graph Minors, Treewidth, Graph algorithms, Parameterized Algorithms} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9511, Parameterized complexity and approximation algorithms (2010)

From 14. 12. 2009 to 17. 12. 2009., the Dagstuhl Seminar 09511
``Parameterized complexity and approximation algorithms '' was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dániel Marx. 09511 Abstracts Collection – Parameterized complexity and approximation algorithms. In Parameterized complexity and approximation algorithms. Dagstuhl Seminar Proceedings, Volume 9511, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{demaine_et_al:DagSemProc.09511.1, author = {Demaine, Erik D. and Hajiaghayi, MohammadTaghi and Marx, D\'{a}niel}, title = {{09511 Abstracts Collection – Parameterized complexity and approximation algorithms}}, booktitle = {Parameterized complexity and approximation algorithms}, pages = {1--14}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9511}, editor = {Erik D. Demaine and MohammadTaghi Hajiaghayi and D\'{a}niel Marx}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09511.1}, URN = {urn:nbn:de:0030-drops-25025}, doi = {10.4230/DagSemProc.09511.1}, annote = {Keywords: Parameterized complexity, Approximation algorithms} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9511, Parameterized complexity and approximation algorithms (2010)

Many of the computational problems that arise in practice are optimization
problems: the task is to find a solution where the cost, quality, size,
profit, or some other measure is as large or small as possible. The
NP-hardness of an optimization problem implies that, unless P = NP, there is
no polynomial-time algorithm that finds the exact value of the optimum.
Various approaches have been proposed in the literature to cope with NP-hard
problems. When designing approximation algorithms, we relax the requirement
that the algorithm produces an optimum solution, and our aim is to devise a
polynomial-time algorithm such that the solution it produces is not
necessarily optimal, but there is some worst-case bound on the solution
quality.

Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dániel Marx. 09511 Executive Summary – Parameterized complexity and approximation algorithms. In Parameterized complexity and approximation algorithms. Dagstuhl Seminar Proceedings, Volume 9511, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{demaine_et_al:DagSemProc.09511.2, author = {Demaine, Erik D. and Hajiaghayi, MohammadTaghi and Marx, D\'{a}niel}, title = {{09511 Executive Summary – Parameterized complexity and approximation algorithms}}, booktitle = {Parameterized complexity and approximation algorithms}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9511}, editor = {Erik D. Demaine and MohammadTaghi Hajiaghayi and D\'{a}niel Marx}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09511.2}, URN = {urn:nbn:de:0030-drops-25011}, doi = {10.4230/DagSemProc.09511.2}, annote = {Keywords: Parameterized complexity, Approximation algorithms} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9511, Parameterized complexity and approximation algorithms (2010)

The paper contains a list of the problems presented on Monday, December 14, 2009 at the open problem session of the Seminar on Parameterized Complexity and Approximation Algorithms, held at Schloss Dagstuhl in Wadern, Germany.

Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dániel Marx. 09511 Open Problems – Parameterized complexity and approximation algorithms. In Parameterized complexity and approximation algorithms. Dagstuhl Seminar Proceedings, Volume 9511, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{demaine_et_al:DagSemProc.09511.3, author = {Demaine, Erik D. and Hajiaghayi, MohammadTaghi and Marx, D\'{a}niel}, title = {{09511 Open Problems – Parameterized complexity and approximation algorithms}}, booktitle = {Parameterized complexity and approximation algorithms}, pages = {1--10}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9511}, editor = {Erik D. Demaine and MohammadTaghi Hajiaghayi and D\'{a}niel Marx}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09511.3}, URN = {urn:nbn:de:0030-drops-24992}, doi = {10.4230/DagSemProc.09511.3}, annote = {Keywords: Parameterized complexity, approximation algorithms, open problems} }

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

We analyze the structure of equilibria and the price of anarchy in the family of network creation games considered extensively in the past few years, which attempt to unify the network design and network routing problems by modeling both creation and usage costs. In general, the games are played on a host graph, where each node is a selfish independent agent (player) and each edge has a fixed link creation cost~$\alpha$. Together the agents create a network (a subgraph of the host graph) while selfishly minimizing the link creation costs plus the sum of the distances to all other players (usage cost). In this paper, we pursue two important facets of the network creation~game.
First, we study extensively a natural version of the game, called the cooperative model, where nodes can collaborate and share the cost of creating any edge in the host graph. We prove the first nontrivial bounds in this model, establishing that the price of anarchy is polylogarithmic in $n$ for all values of~$\alpha$ in complete host graphs. This bound is the first result of this type for any version of the network creation game; most previous general upper bounds are polynomial in~$n$. Interestingly, we also show that equilibrium graphs have polylogarithmic diameter for the most natural range of~$\alpha$ (at most $n \mathop{\rm polylg}\nolimits n$).
Second, we study the impact of the natural assumption that the host graph is a general graph, not necessarily complete. This model is a simple example of nonuniform creation costs among the edges (effectively allowing weights of $\alpha$ and~$\infty$). We prove the first assemblage of upper and lower bounds for this context, establishing nontrivial tight bounds for many ranges of~$\alpha$, for both the unilateral and cooperative versions of network creation. In particular, we establish polynomial lower bounds for both versions and many ranges of~$\alpha$, even for this simple nonuniform cost model, which sharply contrasts the conjectured constant bounds for these games in complete (uniform) graphs.

Erik D. Demaine, MohammadTaghi Hajiaghayi, Hamid Mahini, and Morteza Zadimoghaddam. The Price of Anarchy in Cooperative Network Creation Games. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 301-312, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{demaine_et_al:LIPIcs.STACS.2009.1839, author = {Demaine, Erik D. and Hajiaghayi, MohammadTaghi and Mahini, Hamid and Zadimoghaddam, Morteza}, title = {{The Price of Anarchy in Cooperative Network Creation Games}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {301--312}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1839}, URN = {urn:nbn:de:0030-drops-18390}, doi = {10.4230/LIPIcs.STACS.2009.1839}, annote = {Keywords: } }

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