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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

We study single-machine scheduling problems with few deadlines. We focus on two classical objectives, namely minimizing the weighted number of tardy jobs and the total weighted completion time. For both problems, we give a pseudopolynomial-time algorithm for a constant number of different deadlines. This algorithm is complemented with an ETH-based, almost tight lower bound. Furthermore, we study the case where the number of jobs with a nontrivial deadline is taken as parameter. For this case, the complexity of our two problems differ: Minimizing the total number of tardy jobs becomes fixed-parameter tractable, while minimizing the total weighted completion time is W[1]-hard.

Klaus Heeger, Danny Hermelin, and Dvir Shabtay. Single Machine Scheduling with Few Deadlines. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{heeger_et_al:LIPIcs.IPEC.2023.24, author = {Heeger, Klaus and Hermelin, Danny and Shabtay, Dvir}, title = {{Single Machine Scheduling with Few Deadlines}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.24}, URN = {urn:nbn:de:0030-drops-194434}, doi = {10.4230/LIPIcs.IPEC.2023.24}, annote = {Keywords: Single-machine scheduling, weighted completion time, tardy jobs, pseudopolynomial algorithms, parameterized complexity} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

We provide new (parameterized) computational hardness results for Interval Scheduling on Unrelated Machines. It is a classical scheduling problem motivated from just-in-time or lean manufacturing, where the goal is to complete jobs exactly at their deadline. We are given n jobs and m machines. Each job has a deadline, a weight, and a processing time that may be different on each machine. The goal is find a schedule that maximizes the total weight of jobs completed exactly at their deadline. Note that this uniquely defines a processing time interval for each job on each machine.
Interval Scheduling on Unrelated Machines is closely related to coloring interval graphs and has been thoroughly studied for several decades. However, as pointed out by Mnich and van Bevern [Computers & Operations Research, 2018], the parameterized complexity for the number m of machines as a parameter remained open. We resolve this by showing that Interval Scheduling on Unrelated Machines is W[1]-hard when parameterized by the number m of machines. To this end, we prove W[1]-hardness with respect to m of the special case where we have parallel machines with eligible machine sets for jobs. This answers Open Problem 8 of Mnich and van Bevern’s list of 15 open problems in the parameterized complexity of scheduling [Computers & Operations Research, 2018].
Furthermore, we resolve the computational complexity status of the unweighted version of Interval Scheduling on Unrelated Machines by proving that it is NP-complete. This answers an open question by Sung and Vlach [Journal of Scheduling, 2005].

Danny Hermelin, Yuval Itzhaki, Hendrik Molter, and Dvir Shabtay. Hardness of Interval Scheduling on Unrelated Machines. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hermelin_et_al:LIPIcs.IPEC.2022.18, author = {Hermelin, Danny and Itzhaki, Yuval and Molter, Hendrik and Shabtay, Dvir}, title = {{Hardness of Interval Scheduling on Unrelated Machines}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.18}, URN = {urn:nbn:de:0030-drops-173748}, doi = {10.4230/LIPIcs.IPEC.2022.18}, annote = {Keywords: Just-in-time scheduling, Parallel machines, Eligible machine sets, W\lbrack1\rbrack-hardness, NP-hardness} }

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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)

Given N instances (X_1,t_1),…,(X_N,t_N) of Subset Sum, the AND Subset Sum problem asks to determine whether all of these instances are yes-instances; that is, whether each set of integers X_i has a subset that sums up to the target integer t_i. We prove that this problem cannot be solved in time Õ((N ⋅ t_max)^{1-ε}), for t_max = max_i t_i and any ε > 0, assuming the ∀ ∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude Õ(n+P_max⋅n^{1-ε})-time algorithms for several scheduling problems on n jobs with maximum processing time P_max, assuming ∀∃-SETH. These include classical problems such as 1||∑ w_jU_j, the problem of minimizing the total weight of tardy jobs on a single machine, and P₂||∑ U_j, the problem of minimizing the number of tardy jobs on two identical parallel machines.

Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. Scheduling Lower Bounds via AND Subset Sum. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{abboud_et_al:LIPIcs.ICALP.2020.4, author = {Abboud, Amir and Bringmann, Karl and Hermelin, Danny and Shabtay, Dvir}, title = {{Scheduling Lower Bounds via AND Subset Sum}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {4:1--4:15}, 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.2020.4}, URN = {urn:nbn:de:0030-drops-124119}, doi = {10.4230/LIPIcs.ICALP.2020.4}, annote = {Keywords: SETH, fine grained complexity, Subset Sum, scheduling} }

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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)

This paper is concerned with the 1||∑ p_jU_j problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also a very important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs in O(P ⋅ n) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date.
In this paper we develop two new algorithms for 1||∑ p_jU_j, each improving on Lawler and Moore’s algorithm in a different scenario:
- Our first algorithm runs in Õ(P^{7/4}) time, and outperforms Lawler and Moore’s algorithm in instances where n = ω̃(P^{3/4}).
- Our second algorithm runs in Õ(min{P ⋅ D_#, P + D}) time, where D_# is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore’s algorithm when n = ω̃(D_#) or n = ω̃(D/P). Further, it extends the known Õ(P) algorithm for the single due date special case of 1||∑ p_jU_j in a natural way.
Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, while for the first algorithm we define a new "skewed" version of (max,min)-convolution which is interesting in its own right.

Karl Bringmann, Nick Fischer, Danny Hermelin, Dvir Shabtay, and Philip Wellnitz. Faster Minimization of Tardy Processing Time on a Single Machine. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 19:1-19:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bringmann_et_al:LIPIcs.ICALP.2020.19, author = {Bringmann, Karl and Fischer, Nick and Hermelin, Danny and Shabtay, Dvir and Wellnitz, Philip}, title = {{Faster Minimization of Tardy Processing Time on a Single Machine}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {19:1--19:12}, 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.2020.19}, URN = {urn:nbn:de:0030-drops-124269}, doi = {10.4230/LIPIcs.ICALP.2020.19}, annote = {Keywords: Weighted number of tardy jobs, sumsets, convolutions} }

Document

**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

We study a scheduling problem where two agents (each equipped with a private set of jobs) compete to perform their respective jobs on a common single machine. Each agent wants to keep the weighted sum of completion times of his jobs below a given (agent-dependent) bound. This problem is known to be NP-hard, even for quite restrictive settings of the problem parameters.
We consider parameterized versions of the problem where one of the agents has a small number of jobs (and where this small number constitutes the parameter). The problem becomes much more tangible in this case, and we present three positive algorithmic results for it. Our study is complemented by showing that the general problem is NP-complete even when one agent only has a single job.

Danny Hermelin, Judith-Madeleine Kubitza, Dvir Shabtay, Nimrod Talmon, and Gerhard Woeginger. Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 55-65, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hermelin_et_al:LIPIcs.IPEC.2015.55, author = {Hermelin, Danny and Kubitza, Judith-Madeleine and Shabtay, Dvir and Talmon, Nimrod and Woeginger, Gerhard}, title = {{Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {55--65}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.55}, URN = {urn:nbn:de:0030-drops-55713}, doi = {10.4230/LIPIcs.IPEC.2015.55}, annote = {Keywords: Parameterized Complexity, Multiagent Scheduling} }

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