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

We develop a framework for obtaining polynomial time approximation schemes (PTAS) for a class of stochastic dynamic programs. Using our framework, we obtain the first PTAS for the following stochastic combinatorial optimization problems:
1) Probemax [Munagala, 2016]: We are given a set of n items, each item i in [n] has a value X_i which is an independent random variable with a known (discrete) distribution pi_i. We can probe a subset P subseteq [n] of items sequentially. Each time after {probing} an item i, we observe its value realization, which follows the distribution pi_i. We can adaptively probe at most m items and each item can be probed at most once. The reward is the maximum among the m realized values. Our goal is to design an adaptive probing policy such that the expected value of the reward is maximized. To the best of our knowledge, the best known approximation ratio is 1-1/e, due to Asadpour et al. [Asadpour and Nazerzadeh, 2015]. We also obtain PTAS for some generalizations and variants of the problem.
2) Committed Pandora's Box [Weitzman, 1979; Singla, 2018]: We are given a set of n boxes. For each box i in [n], the cost c_i is deterministic and the value X_i is an independent random variable with a known (discrete) distribution pi_i. Opening a box i incurs a cost of c_i. We can adaptively choose to open the boxes (and observe their values) or stop. We want to maximize the expectation of the realized value of the last opened box minus the total opening cost.
3) Stochastic Target [{I}lhan et al., 2011]: Given a predetermined target T and n items, we can adaptively insert the items into a knapsack and insert at most m items. Each item i has a value X_i which is an independent random variable with a known (discrete) distribution. Our goal is to design an adaptive policy such that the probability of the total values of all items inserted being larger than or equal to T is maximized. We provide the first bi-criteria PTAS for the problem.
4) Stochastic Blackjack Knapsack [Levin and Vainer, 2014]: We are given a knapsack of capacity C and probability distributions of n independent random variables X_i. Each item i in [n] has a size X_i and a profit p_i. We can adaptively insert the items into a knapsack, as long as the capacity constraint is not violated. We want to maximize the expected total profit of all inserted items. If the capacity constraint is violated, we lose all the profit. We provide the first bi-criteria PTAS for the problem.

Hao Fu, Jian Li, and Pan Xu. A PTAS for a Class of Stochastic Dynamic Programs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fu_et_al:LIPIcs.ICALP.2018.56, author = {Fu, Hao and Li, Jian and Xu, Pan}, title = {{A PTAS for a Class of Stochastic Dynamic Programs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {56:1--56:14}, 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.2018.56}, URN = {urn:nbn:de:0030-drops-90609}, doi = {10.4230/LIPIcs.ICALP.2018.56}, annote = {Keywords: stochastic optimization, dynamic program, markov decision process, block policy, approximation algorithm} }

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

Online matching has received significant attention over the last 15 years due to its close connection to Internet advertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal (1 - 1/epsilon) competitive ratio in the standard adversarial online model, much effort has gone into developing useful online models that incorporate some stochasticity in the arrival process. One such popular model is the "known I.I.D. model" where different customer-types arrive online from a known distribution. We develop algorithms with improved competitive ratios for some basic variants of this model with integral arrival rates, including: (a) the case of general weighted edges, where we improve the best-known ratio of 0.667 due to [Haeupler, Mirrokni and Zadimoghaddam WINE 2011] to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of [Jaillet and Lu Math. Oper. Res 2013] to 0.7299. We also consider two extensions, one is "known I.I.D." with non-integral arrival rate and stochastic rewards; the other is "known I.I.D." b-matching with non-integral arrival rate and stochastic rewards. We present a simple non-adaptive algorithm which works well simultaneously on the two extensions.
One of the key ingredients of our improvement is the following (offline) approach to bipartite-matching polytopes with additional constraints. We first add several valid constraints in order to get a good fractional solution f; however, these give us less control over the structure of f. We next remove all these additional constraints and randomly move from f to a feasible point on the matching polytope with all coordinates being from the set {0, 1/k, 2/k,..., 1} for a chosen integer k. The structure of this solution is inspired by [Jaillet and Lu Math. Oper. Res 2013] and is a tractable structure for algorithm design and analysis. The appropriate random move preserves many of the removed constraints (approximately [exactly] with high probability [in expectation]). This underlies some of our improvements, and, we hope, could be of independent interest.

Brian Brubach, Karthik Abinav Sankararaman, Aravind Srinivasan, and Pan Xu. New Algorithms, Better Bounds, and a Novel Model for Online Stochastic Matching. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{brubach_et_al:LIPIcs.ESA.2016.24, author = {Brubach, Brian and Sankararaman, Karthik Abinav and Srinivasan, Aravind and Xu, Pan}, title = {{New Algorithms, Better Bounds, and a Novel Model for Online Stochastic Matching}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {24:1--24:16}, 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.2016.24}, URN = {urn:nbn:de:0030-drops-63753}, doi = {10.4230/LIPIcs.ESA.2016.24}, annote = {Keywords: Ad-Allocation, Online Matching, Randomized Algorithms} }

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

We consider two fundamental problems in stochastic optimization: approximation algorithms for stochastic matching, and sampling bounds in the black-box model. For the former, we improve the current-best bound of 3.709 due to Adamczyk et al. (2015), to 3.224; we also present improvements on Bansal et al. (2012) for hypergraph matching and for relaxed versions of the problem. In the context of stochastic optimization, we improve upon the sampling bounds of Charikar et al. (2005).

Alok Baveja, Amit Chavan, Andrei Nikiforov, Aravind Srinivasan, and Pan Xu. Improved Bounds in Stochastic Matching and Optimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 124-134, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{baveja_et_al:LIPIcs.APPROX-RANDOM.2015.124, author = {Baveja, Alok and Chavan, Amit and Nikiforov, Andrei and Srinivasan, Aravind and Xu, Pan}, title = {{Improved Bounds in Stochastic Matching and Optimization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {124--134}, 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.2015.124}, URN = {urn:nbn:de:0030-drops-52991}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.124}, annote = {Keywords: stochastic matching, approximation algorithms, sampling complexity} }