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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We study a bilevel optimization problem which is a zero-sum Stackelberg game. In this problem, there are two players, a leader and a follower, who pick items from a common set. Both the leader and the follower have their own (multi-dimensional) budgets, respectively. Each item is associated with a profit, which is the same to the leader and the follower, and will consume the leader’s (follower’s) budget if it is selected by the leader (follower). The leader and the follower will select items in a sequential way: First, the leader selects items within the leader’s budget. Then the follower selects items from the remaining items within the follower’s budget. The goal of the leader is to minimize the maximum profit that the follower can obtain. Let s_A and s_B be the dimension of the leader’s and follower’s budget, respectively. A special case of our problem is the bilevel knapsack problem studied by Caprara et al. [SIAM Journal on Optimization, 2014], where s_A = s_B = 1. We consider the general problem and obtain an (s_B+ε)-approximation algorithm when s_A and s_B are both constant. In particular, if s_B = 1, our algorithm implies a PTAS for the bilevel knapsack problem, which is the first 𝒪(1)-approximation algorithm. We also complement our result by showing that there does not exist any (4/3-ε)-approximation algorithm even if s_A = 1 and s_B = 2. We also consider a variant of our problem with resource augmentation when s_A and s_B are both part of the input. We obtain an 𝒪(1)-approximation algorithm with 𝒪(1)-resource augmentation, that is, we give an algorithm that returns a solution which exceeds the given leader’s budget by 𝒪(1) times, and the objective value achieved by the solution is 𝒪(1) times the optimal objective value that respects the leader’s budget.

Lin Chen, Xiaoyu Wu, and Guochuan Zhang. Approximation Algorithms for Interdiction Problem with Packing Constraints. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2022.39, author = {Chen, Lin and Wu, Xiaoyu and Zhang, Guochuan}, title = {{Approximation Algorithms for Interdiction Problem with Packing Constraints}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {39:1--39:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.39}, URN = {urn:nbn:de:0030-drops-163806}, doi = {10.4230/LIPIcs.ICALP.2022.39}, annote = {Keywords: Bilevel Integer Programming, Interdiction Constraints, Knapsack} }

Document

Complete Volume

**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

LIPIcs, Volume 149, ISAAC'19, Complete Volume

30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{lu_et_al:LIPIcs.ISAAC.2019, title = {{LIPIcs, Volume 149, ISAAC'19, Complete Volume}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019}, URN = {urn:nbn:de:0030-drops-116417}, doi = {10.4230/LIPIcs.ISAAC.2019}, annote = {Keywords: Theory of computation; Models of computation; Computational complexity and cryptography; Randomness, geometry and discrete structures; Theory and algorithms for application domains; Design and analysis of algorithms} }

Document

Front Matter

**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Front Matter, Table of Contents, Preface, Symposium Organization

30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lu_et_al:LIPIcs.ISAAC.2019.0, author = {Lu, Pinyan and Zhang, Guochuan}, title = {{Front Matter, Table of Contents, Preface, Symposium Organization}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.0}, URN = {urn:nbn:de:0030-drops-114967}, doi = {10.4230/LIPIcs.ISAAC.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Symposium Organization} }

Document

**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We study approximation and parameterized algorithms for R||C_max, focusing on the problem when the rank of the matrix formed by job processing times is small. Bhaskara et al. initiated the study of approximation algorithms with respect to the rank, showing that R||C_max admits a QPTAS (Quasi-polynomial time approximation scheme) when the rank is 2, and becomes APX-hard when the rank is 4.
We continue this line of research. We prove that R||C_max is APX-hard even if the rank is 3, resolving an open problem. We then show that R||C_max is FPT parameterized by the rank and the largest job processing time p_max. This generalizes the parameterized results on P||C_max and R||C_max with few different types of machines. We also provide nearly tight lower bounds under Exponential Time Hypothesis which suggests that the running time of the FPT algorithm is unlikely to be improved significantly.

Lin Chen, Dániel Marx, Deshi Ye, and Guochuan Zhang. Parameterized and Approximation Results for Scheduling with a Low Rank Processing Time Matrix. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.STACS.2017.22, author = {Chen, Lin and Marx, D\'{a}niel and Ye, Deshi and Zhang, Guochuan}, title = {{Parameterized and Approximation Results for Scheduling with a Low Rank Processing Time Matrix}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.22}, URN = {urn:nbn:de:0030-drops-70110}, doi = {10.4230/LIPIcs.STACS.2017.22}, annote = {Keywords: APX-hardness, Parameterized algorithm, Scheduling, Exponential Time Hypothesis} }

Document

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

We consider the problem of scheduling with renewable speed-up resources. Given m identical machines, n jobs and c different discrete resources, the task is to schedule each job non-preemptively onto one of the machines so as to minimize the makespan. In our problem, a job has its original processing time, which could be reduced by utilizing one of the resources. As resources are different, the amount of the time reduced for each job is different depending on the resource it uses. Once a resource is being used by one job, it can not be used simultaneously by any other job until this job is finished, hence the scheduler should take into account the job-to-machine assignment together with the resource-to-job assignment.
We observe that, the classical unrelated machine scheduling problem is actually a special case of our problem when m=c, i.e., the number of resources equals the number of machines. Extending the techniques for the unrelated machine scheduling, we give a 2-approximation algorithm when both m and c are part of the input. We then consider two special cases for the problem, with m or c being a constant, and derive PTASes (Polynomial Time Approximation Schemes) respectively. We also establish the relationship between the two parameters m and c, through which we are able to transform the PTAS for the case when m is constant to the case when c is a constant. The relationship between the two parameters reveals the structure within the problem, and may be of independent interest.

Lin Chen, Deshi Ye, and Guochuan Zhang. Approximation Algorithms for Parallel Machine Scheduling with Speed-up Resources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2016.5, author = {Chen, Lin and Ye, Deshi and Zhang, Guochuan}, title = {{Approximation Algorithms for Parallel Machine Scheduling with Speed-up Resources}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {5:1--5:12}, 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.2016.5}, URN = {urn:nbn:de:0030-drops-66283}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.5}, annote = {Keywords: approximation algorithms, scheduling, linear programming} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We consider a natural generalization of the classical multiple knapsack problem in which instead of packing single items we are packing groups of items. In this problem, we have multiple knapsacks and a set of items which are partitioned into groups. Each item has an individual weight, while the profit is associated with groups rather than items. The profit of a group can be attained if and only if every item of this group is packed. Such a general model finds applications in various practical problems, e.g., delivering bundles of goods. The tractability of this problem relies heavily on how large a group could be. Deciding if a group of items of total weight 2 could be packed into two knapsacks of unit capacity is already NP-hard and it thus rules out a constant-approximation algorithm for this problem in general. We then focus on the parameterized version where the total weight of items in each group is bounded by a factor delta of the total capacity of all knapsacks. Both approximation and inapproximability results with respect to delta are derived. We also show that, depending on whether the number of knapsacks is a constant or part of the input, the approximation ratio for the problem, as a function on delta, changes substantially, which has a clear difference from the classical multiple knapsack problem.

Lin Chen and Guochuan Zhang. Packing Groups of Items into Multiple Knapsacks. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chen_et_al:LIPIcs.STACS.2016.28, author = {Chen, Lin and Zhang, Guochuan}, title = {{Packing Groups of Items into Multiple Knapsacks}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.28}, URN = {urn:nbn:de:0030-drops-57299}, doi = {10.4230/LIPIcs.STACS.2016.28}, annote = {Keywords: approximation algorithms, lower bound, multiple knapsack, bin packing} }

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