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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

We consider the online makespan minimization problem on identical machines. Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT) algorithm is 1.5-competitive. For the special case of two machines, Noga and Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive ratio of (5 - sqrt{5})/2 ~~ 1.382, matching the lower bound by Chen and Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many applications one can kill a job and restart it later, and they leave an open problem whether algorithms with restart can obtain better competitive ratios.
We resolve this long-standing open problem on the positive end. Our algorithm has a natural rule for killing a processing job: a newly-arrived job replaces the smallest processing job if 1) the new job is larger than other pending jobs, 2) the new job is much larger than the processing one, and 3) the processed portion is small relative to the size of the new job. With appropriate choice of parameters, we show that our algorithm improves the 1.5 competitive ratio for the general case, and the 1.382 competitive ratio for the two-machine case.

Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Online Makespan Minimization: The Power of Restart. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{huang_et_al:LIPIcs.APPROX-RANDOM.2018.14, author = {Huang, Zhiyi and Kang, Ning and Tang, Zhihao Gavin and Wu, Xiaowei and Zhang, Yuhao}, title = {{Online Makespan Minimization: The Power of Restart}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {14:1--14:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.14}, URN = {urn:nbn:de:0030-drops-94182}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.14}, annote = {Keywords: Online Scheduling, Makespan Minimization, Identical Machines} }

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

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.

Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{huang_et_al:LIPIcs.ICALP.2018.79, author = {Huang, Zhiyi and Tang, Zhihao Gavin and Wu, Xiaowei and Zhang, Yuhao}, title = {{Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {79:1--79:14}, 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.2018.79}, URN = {urn:nbn:de:0030-drops-90830}, doi = {10.4230/LIPIcs.ICALP.2018.79}, annote = {Keywords: Vertex Weighted, Online Bipartite Matching, Randomized Primal-Dual} }

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

We consider the online vector packing problem in which we have a d dimensional knapsack and items u with weight vectors w_u in R_+^d arrive online in an arbitrary order. Upon the arrival of an item, the algorithm must decide immediately whether to discard or accept the item into the knapsack. When item u is accepted, w_u(i) units of capacity on dimension i will be taken up, for each i in [d]. To satisfy the knapsack constraint, an accepted item can be later disposed of with no cost, but discarded or disposed of items cannot be recovered. The objective is to maximize the utility of the accepted items S at the end of the algorithm, which is given by f(S) for some non-negative monotone submodular function f.
For any small constant epsilon > 0, we consider the special case that the weight of an item on every dimension is at most a (1- epsilon) fraction of the total capacity, and give a polynomial-time deterministic O(k / epsilon^2)-competitive algorithm for the problem, where k is the (column) sparsity of the weight vectors. We also show several (almost) tight hardness results even when the algorithm is computationally unbounded. We first show that under the epsilon-slack assumption, no deterministic algorithm can obtain any o(k) competitive ratio, and no randomized algorithm can obtain any o(k / log k) competitive ratio. We then show that for the general case (when epsilon = 0), no randomized algorithm can obtain any o(k) competitive ratio.
In contrast to the (1+delta) competitive ratio achieved in Kesselheim et al. [STOC 2014] for the problem with random arrival order of items and under large capacity assumption, we show that in the arbitrary arrival order case, even when |w_u|_infinity is arbitrarily small for all items u, it is impossible to achieve any o(log k / log log k) competitive ratio.

T.-H. Hubert Chan, Shaofeng H.-C. Jiang, Zhihao Gavin Tang, and Xiaowei Wu. Online Submodular Maximization Problem with Vector Packing Constraint. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chan_et_al:LIPIcs.ESA.2017.24, author = {Chan, T.-H. Hubert and Jiang, Shaofeng H.-C. and Tang, Zhihao Gavin and Wu, Xiaowei}, title = {{Online Submodular Maximization Problem with Vector Packing Constraint}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {24:1--24:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.24}, URN = {urn:nbn:de:0030-drops-78190}, doi = {10.4230/LIPIcs.ESA.2017.24}, annote = {Keywords: Submodular Maximization, Free-disposal, Vector Packing} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

We study the max-min fair allocation problem in which a set of m indivisible items are to be distributed among n agents such that the minimum utility among all agents is maximized. In the restricted setting, the utility of each item j on agent i is either 0 or some non-negative weight w_j. For this setting, Asadpour et al. [TALG, 2012] showed that a certain configuration-LP can be used to estimate the optimal value within a factor of 4 + delta, for any delta > 0, which was recently extended by Annamalai et al. [SODA 2015] to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani [SIGecom Exch., 2005] showed that it is NP-hard to approximate the problem within any ratio smaller than 2.
In this paper we consider the (1, epsilon)-restricted max-min fair allocation problem, in which for some parameter epsilon in (0, 1), each item j is either heavy (w_j = 1) or light (w_j = epsilon). We show that the (1, epsilon)-restricted case is also NP-hard to approximate within any ratio smaller than 2. Hence, this simple special case is still algorithmically interesting.
Using the configuration-LP, we are able to estimate the optimal value of the problem within a factor of 3 + delta, for any delta > 0. Extending this idea, we also obtain a quasi-polynomial time (3 + 4 epsilon)-approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as epsilon tends to 0, the approximation ratio of our polynomial-time algorithm approaches 3 + 2 sqrt{2} approx 5.83.

T-H. Hubert Chan, Zhihao Gavin Tang, and Xiaowei Wu. On (1, epsilon)-Restricted Max-Min Fair Allocation Problem. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chan_et_al:LIPIcs.ISAAC.2016.23, author = {Chan, T-H. Hubert and Tang, Zhihao Gavin and Wu, Xiaowei}, title = {{On (1, epsilon)-Restricted Max-Min Fair Allocation Problem}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {23:1--23:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.23}, URN = {urn:nbn:de:0030-drops-67939}, doi = {10.4230/LIPIcs.ISAAC.2016.23}, annote = {Keywords: Max-Min Fair Allocation, Hypergraph Matching} }

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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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