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**Published in:** LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Datalog^∘ is an extension of Datalog, where instead of a program being a collection of union of conjunctive queries over the standard Boolean semiring, a program may now be a collection of sum-product queries over an arbitrary commutative partially ordered pre-semiring. Datalog^∘ is more powerful than Datalog in that its additional algebraic structure alows for supporting recursion with aggregation. At the same time, Datalog^∘ retains the syntactic and semantic simplicity of Datalog: Datalog^∘ has declarative least fixpoint semantics. The least fixpoint can be found via the naïve evaluation algorithm that repeatedly applies the immediate consequence operator until no further change is possible.
It was shown in [Mahmoud Abo Khamis et al., 2022] that, when the underlying semiring is p-stable, then the naïve evaluation of any Datalog^∘ program over the semiring converges in a finite number of steps. However, the upper bounds on the rate of convergence were exponential in the number n of ground IDB atoms.
This paper establishes polynomial upper bounds on the convergence rate of the naïve algorithm on linear Datalog^∘ programs, which is quite common in practice. In particular, the main result of this paper is that the convergence rate of linear Datalog^∘ programs under any p-stable semiring is O(pn³). Furthermore, we show a matching lower bound by constructing a p-stable semiring and a linear Datalog^∘ program that requires Ω(pn³) iterations for the naïve iteration algorithm to converge. Next, we study the convergence rate in terms of the number of elements in the semiring for linear Datalog^∘ programs. When L is the number of elements, the convergence rate is bounded by O(pn log L). This significantly improves the convergence rate for small L. We show a nearly matching lower bound as well.

Sungjin Im, Benjamin Moseley, Hung Ngo, and Kirk Pruhs. On the Convergence Rate of Linear Datalog ^∘ over Stable Semirings. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{im_et_al:LIPIcs.ICDT.2024.11, author = {Im, Sungjin and Moseley, Benjamin and Ngo, Hung and Pruhs, Kirk}, title = {{On the Convergence Rate of Linear Datalog ^∘ over Stable Semirings}}, booktitle = {27th International Conference on Database Theory (ICDT 2024)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-312-6}, ISSN = {1868-8969}, year = {2024}, volume = {290}, editor = {Cormode, Graham and Shekelyan, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.11}, URN = {urn:nbn:de:0030-drops-197939}, doi = {10.4230/LIPIcs.ICDT.2024.11}, annote = {Keywords: Datalog, convergence rate, semiring} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We consider the classic online learning and stochastic multi-armed bandit (MAB) problems, when at each step, the online policy can probe and find out which of a small number (k) of choices has better reward (or loss) before making its choice. In this model, we derive algorithms whose regret bounds have exponentially better dependence on the time horizon compared to the classic regret bounds. In particular, we show that probing with k = 2 suffices to achieve time-independent regret bounds for online linear and convex optimization. The same number of probes improve the regret bound of stochastic MAB with independent arms from O(√{nT}) to O(n² log T), where n is the number of arms and T is the horizon length. For stochastic MAB, we also consider a stronger model where a probe reveals the reward values of the probed arms, and show that in this case, k = 3 probes suffice to achieve parameter-independent constant regret, O(n²). Such regret bounds cannot be achieved even with full feedback after the play, showcasing the power of limited "advice" via probing before making the play. We also present extensions to the setting where the hints can be imperfect, and to the case of stochastic MAB where the rewards of the arms can be correlated.

Aditya Bhaskara, Sreenivas Gollapudi, Sungjin Im, Kostas Kollias, and Kamesh Munagala. Online Learning and Bandits with Queried Hints. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 16:1-16:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bhaskara_et_al:LIPIcs.ITCS.2023.16, author = {Bhaskara, Aditya and Gollapudi, Sreenivas and Im, Sungjin and Kollias, Kostas and Munagala, Kamesh}, title = {{Online Learning and Bandits with Queried Hints}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {16:1--16:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.16}, URN = {urn:nbn:de:0030-drops-175197}, doi = {10.4230/LIPIcs.ITCS.2023.16}, annote = {Keywords: Online learning, multi-armed bandits, regret} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We consider the Matrix Tree Multiplication problem. This problem is a generalization of the classic Matrix Chain Multiplication problem covered in the dynamic programming chapter of many introductory algorithms textbooks. An instance of the Matrix Tree Multiplication problem consists of a rooted tree with a matrix associated with each edge. The output is, for each leaf in the tree, the product of the matrices on the chain/path from the root to that leaf. Matrix multiplications that are shared between various chains need only be computed once, potentially being shared between different root to leaf chains. Algorithms are evaluated by the number of scalar multiplications performed. Our main result is a linear time algorithm for which the number of scalar multiplications performed is at most 15 times the optimal number of scalar multiplications.

Mahmoud Abo-Khamis, Ryan Curtin, Sungjin Im, Benjamin Moseley, Hung Ngo, Kirk Pruhs, and Alireza Samadian. An Approximation Algorithm for the Matrix Tree Multiplication Problem. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{abokhamis_et_al:LIPIcs.MFCS.2021.6, author = {Abo-Khamis, Mahmoud and Curtin, Ryan and Im, Sungjin and Moseley, Benjamin and Ngo, Hung and Pruhs, Kirk and Samadian, Alireza}, title = {{An Approximation Algorithm for the Matrix Tree Multiplication Problem}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.6}, URN = {urn:nbn:de:0030-drops-144464}, doi = {10.4230/LIPIcs.MFCS.2021.6}, annote = {Keywords: Matrix Multiplication, Approximation Algorithm} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d, this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O((log d)/(log log d)) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper:
- We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8/3 for two dimensions.
- If the value of opt is known, we improve the competitive ratio to 9/4 using a variant of the classic best fit algorithm for two dimensions.
- For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem.

Ilan Cohen, Sungjin Im, and Debmalya Panigrahi. Online Two-Dimensional Load Balancing. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cohen_et_al:LIPIcs.ICALP.2020.34, author = {Cohen, Ilan and Im, Sungjin and Panigrahi, Debmalya}, title = {{Online Two-Dimensional Load Balancing}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {34:1--34:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.34}, URN = {urn:nbn:de:0030-drops-124415}, doi = {10.4230/LIPIcs.ICALP.2020.34}, annote = {Keywords: Online algorithms, scheduling, multi-dimensional} }

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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the matroid coflow scheduling problem, where each job is comprised of a set of flows and the family of sets that can be scheduled at any time form a matroid. Our main result is a polynomial-time algorithm that yields a 2-approximation for the objective of minimizing the weighted completion time. This result is tight assuming P != NP. As a by-product we also obtain the first (2+epsilon)-approximation algorithm for the preemptive concurrent open shop scheduling problem.

Sungjin Im, Benjamin Moseley, Kirk Pruhs, and Manish Purohit. Matroid Coflow Scheduling. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 145:1-145:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{im_et_al:LIPIcs.ICALP.2019.145, author = {Im, Sungjin and Moseley, Benjamin and Pruhs, Kirk and Purohit, Manish}, title = {{Matroid Coflow Scheduling}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {145:1--145:13}, 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.2019.145}, URN = {urn:nbn:de:0030-drops-107213}, doi = {10.4230/LIPIcs.ICALP.2019.145}, annote = {Keywords: Coflow Scheduling, Concurrent Open Shop, Matroid Scheduling} }

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

We consider a setting where selfish agents want to schedule jobs on related machines. The agent submitting a job picks a server that minimizes a linear combination of the server price and the resulting response time for that job on the selected server. The manager's task is to maintain server prices to (approximately) optimize the maximum response time, which is a measure of social good. We show that the existence of a pricing scheme with certain competitiveness is equivalent to the existence of a monotone immediate-dispatch algorithm. Our main result is a monotone immediate-dispatch algorithm that is O(1)-competitive with respect to the maximum response time.

Sungjin Im, Benjamin Moseley, Kirk Pruhs, and Clifford Stein. Minimizing Maximum Flow Time on Related Machines via Dynamic Posted Pricing. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 51:1-51:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{im_et_al:LIPIcs.ESA.2017.51, author = {Im, Sungjin and Moseley, Benjamin and Pruhs, Kirk and Stein, Clifford}, title = {{Minimizing Maximum Flow Time on Related Machines via Dynamic Posted Pricing}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {51:1--51:10}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.51}, URN = {urn:nbn:de:0030-drops-78287}, doi = {10.4230/LIPIcs.ESA.2017.51}, annote = {Keywords: Posted pricing scheme, online scheduling, related machines, maximum flow time, competitiveness analysis} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Modern data centers consist of a large number of heterogeneous resources such as CPU, memory, network bandwidth, etc. The resources are pooled into clusters for various reasons such as scalability, resource consolidation, and privacy. Clusters are often heterogeneous so that they can better serve jobs with different characteristics submitted from clients. Each job benefits differently depending on how much resource is allocated to the job, which in turn translates to how quickly the job gets completed.
In this paper, we formulate this setting, which we term Multi-Cluster Polytope Scheduling (MCPS). In MCPS, a set of n jobs arrive over time to be executed on m clusters. Each cluster i is associated with a polytope P_i, which constrains how fast one can process jobs assigned to the cluster. For MCPS, we seek to optimize the popular objective of minimizing average weighted flow time of jobs in the online setting. We give a constant competitive algorithm with small constant resource augmentation for a large class of polytopes, which capture many interesting problems that arise in practice. Further, our algorithm is non-clairvoyant. Our algorithm and analysis combine and generalize techniques developed in the recent results for the classical unrelated machines scheduling and the polytope scheduling problem [10,12,11].

Sungjin Im, Janardhan Kulkarni, Benjamin Moseley, and Kamesh Munagala. A Competitive Flow Time Algorithm for Heterogeneous Clusters Under Polytope Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{im_et_al:LIPIcs.APPROX-RANDOM.2016.10, author = {Im, Sungjin and Kulkarni, Janardhan and Moseley, Benjamin and Munagala, Kamesh}, title = {{A Competitive Flow Time Algorithm for Heterogeneous Clusters Under Polytope Constraints}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.10}, URN = {urn:nbn:de:0030-drops-66336}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.10}, annote = {Keywords: Polytope constraints, average flow time, multi-clusters, online scheduling, and competitive analysis} }

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

We consider the classical problem of constrained queueing (or switched networks): There is a set of N queues to which unit sized packets arrive. The queues are interdependent, so that at any time step, only a subset of the queues can be activated. One packet from each activated queue can be transmitted, and leaves the system. The set of feasible subsets that can be activated, denoted S, is downward closed and is known in advance. The goal is to find a scheduling policy that minimizes average delay (or flow time) of the packets. The constrained queueing problem models several practical settings including packet transmission in wireless networks and scheduling cross-bar switches.
In this paper, we study this problem using the the competitive analysis: The packet arrivals can be adversarial and the scheduling policy only uses information about packets currently queued in the system. We present an online algorithm, that for any epsilon > 0, has average flow time at most O(R^2/epsilon^3*OPT+NR) when given (1+epsilon) speed, i.e., the ability to schedule (1+epsilon) packets on average per time step. Here, R is the maximum number of queues that can be simultaneously scheduled, and OPT is the average flow time of the optimal policy. This asymptotic competitive ratio O(R^3/epsilon^3) improves upon the previous O(N/epsilon^2) which was obtained in the context of multi-dimensional scheduling [Im/Kulkarni/Munagala, FOCS 2015]. In the full general model where N can be exponentially larger than R, this is an exponential improvement. The algorithm presented in this paper is based on Makespan estimates which is very different from that in [Im/Kulkarni/Munagala, FOCS 2015], a variation of the Max-Weight algorithm. Further, our policy is myopic, meaning that scheduling decisions at any step are based only on the current composition of the queues. We finally show that speed augmentation is necessary to achieve any bounded competitive ratio.

Sungjin Im, Janardhan Kulkarni, and Kamesh Munagala. Competitive Analysis of Constrained Queueing Systems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 143:1-143:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{im_et_al:LIPIcs.ICALP.2016.143, author = {Im, Sungjin and Kulkarni, Janardhan and Munagala, Kamesh}, title = {{Competitive Analysis of Constrained Queueing Systems}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {143:1--143:13}, 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.2016.143}, URN = {urn:nbn:de:0030-drops-62876}, doi = {10.4230/LIPIcs.ICALP.2016.143}, annote = {Keywords: Online scheduling, Average flow time, Switch network, Adversarial} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We revisit the classical stochastic scheduling problem of nonpreemptively scheduling n jobs so as to minimize total completion time on m identical machines, P \mid \mid \mathbb{E} \sum C_j in the standard 3-field scheduling notation. Previously it was only known how to obtain reasonable approximation if jobs sizes have low variability. However, distributions commonly arising in practice have high variability, and the upper bounds on the approximation ratio for the previous algorithms for such distributions can be even inverse-polynomial in the maximum possible job size. We start by showing that
the natural list scheduling algorithm Shortest Expected Processing Time (SEPT) has a bad approximation ratio for high variability jobs. We observe that a simple randomized rounding of a natural linear programming relaxation is a (1+\epsilon)-machine O(1)-approximation assuming the number of machines is at least logarithmic in the number of jobs. Turning to the case of a modest number of machines, we develop a list scheduling algorithm that is O(\log^2 n + m \log n)-approximate. Our results together imply a (1+\epsilon)-machine O(\log^2 n )-approximation for an arbitrary number of machines. Intuitively our list scheduling algorithm finds an ordering that not only takes the expected size of a job into account, but also takes into account the probability that job will be big.

Sungjin Im, Benjamin Moseley, and Kirk Pruhs. Stochastic Scheduling of Heavy-tailed Jobs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 474-486, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{im_et_al:LIPIcs.STACS.2015.474, author = {Im, Sungjin and Moseley, Benjamin and Pruhs, Kirk}, title = {{Stochastic Scheduling of Heavy-tailed Jobs}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {474--486}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.474}, URN = {urn:nbn:de:0030-drops-49359}, doi = {10.4230/LIPIcs.STACS.2015.474}, annote = {Keywords: stochastic scheduling, completion time, approximation} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

In the (non-preemptive) Generalized Min Sum Set Cover Problem, we
are given n ground elements and a collection of sets S = {S_1,
S_2, ..., S_m} where each set S_i in 2^{[n]} has a positive
requirement k(S_i) that has to be fulfilled. We would like to order all elements to minimize the total (weighted) cover time of all sets. The cover time of a set S_i is defined as the first index j in the ordering such that the first j elements in the ordering contain k(S_i) elements in S_i. This problem was introduced by [Azar, Gamzu and Yin, 2009] with interesting motivations in web page ranking and broadcast scheduling. For this problem, constant approximations are known [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011].
We study the version where preemption is allowed. The difference is
that elements can be fractionally scheduled and a set S is
covered in the moment when k(S) amount of elements in S are scheduled. We give a 2-approximation for this preemptive problem. Our linear programming and analysis are completely different from [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011]. We also show that any preemptive solution can be transformed into a non-preemptive one by losing a factor of 6.2 in the objective function. As a byproduct, we obtain an improved 12.4-approximation for the non-preemptive problem.

Sungjin Im, Maxim Sviridenko, and Ruben van der Zwaan. Preemptive and Non-Preemptive Generalized Min Sum Set Cover. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 465-476, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{im_et_al:LIPIcs.STACS.2012.465, author = {Im, Sungjin and Sviridenko, Maxim and van der Zwaan, Ruben}, title = {{Preemptive and Non-Preemptive Generalized Min Sum Set Cover}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {465--476}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.465}, URN = {urn:nbn:de:0030-drops-33991}, doi = {10.4230/LIPIcs.STACS.2012.465}, annote = {Keywords: Set Cover, Approximation, Preemption, Latency, Average cover time} }