3 Search Results for "Itzhaki, Yuval"

Document
DOI: 10.4230/LIPIcs.STACS.2025.47
Minimizing the Number of Tardy Jobs with Uniform Processing Times on Parallel Machines

Authors: Klaus Heeger and Hendrik Molter

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
In this work, we study the computational (parameterized) complexity of P∣ r_j, p_j = p ∣∑ w_j U_j. Here, we are given m identical parallel machines and n jobs with equal processing time, each characterized by a release date, a due date, and a weight. The task is to find a feasible schedule, that is, an assignment of the jobs to starting times on machines, such that no job starts before its release date and no machine processes several jobs at the same time, that minimizes the weighted number of tardy jobs. A job is considered tardy if it finishes after its due date. Our main contribution is showing that P∣r_j, p_j = p∣∑ U_j (the unweighted version of the problem) is NP-hard and W[2]-hard when parameterized by the number of machines. The former resolves an open problem in Note 2.1.19 by Kravchenko and Werner [Journal of Scheduling, 2011] and Open Problem 2 by Sgall [ESA, 2012], and the latter resolves Open Problem 7 by Mnich and van Bevern [Computers & Operations Research, 2018]. Furthermore, our result shows that the known XP-algorithm by Baptiste et al. [4OR, 2004] for P∣r_j, p_j = p∣∑ w_j U_j parameterized by the number of machines is optimal from a classification standpoint. On the algorithmic side, we provide alternative running time bounds for the above-mentioned known XP-algorithm. Our analysis shows that P∣r_j, p_j = p∣∑ w_j U_j is contained in XP when parameterized by the processing time, and that it is contained in FPT when parameterized by the combination of the number of machines and the processing time. Finally, we give an FPT-algorithm for P∣r_j, p_j = p∣∑ w_j U_j parameterized by the number of release dates or the number of due dates. With this work, we lay out the foundation for a systematic study of the parameterized complexity of P∣r_j, p_j = p∣∑ w_j U_j.
Cite as

Klaus Heeger and Hendrik Molter. Minimizing the Number of Tardy Jobs with Uniform Processing Times on Parallel Machines. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{heeger_et_al:LIPIcs.STACS.2025.47,
  author =	{Heeger, Klaus and Molter, Hendrik},
  title =	{{Minimizing the Number of Tardy Jobs with Uniform Processing Times on Parallel Machines}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{47:1--47:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.47},
  URN =		{urn:nbn:de:0030-drops-228736},
  doi =		{10.4230/LIPIcs.STACS.2025.47},
  annote =	{Keywords: Scheduling, Identical Parallel Machines, Weighted Number of Tardy Jobs, Uniform Processing Times, Release Dates, NP-hard Problems, Parameterized Complexity}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2022.18
Hardness of Interval Scheduling on Unrelated Machines

Authors: Danny Hermelin, Yuval Itzhaki, Hendrik Molter, and Dvir Shabtay

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

Abstract
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].
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SAND.2022.19
Temporal Unit Interval Independent Sets

Authors: Danny Hermelin, Yuval Itzhaki, Hendrik Molter, and Rolf Niedermeier

Published in: LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

Abstract
Temporal graphs have been recently introduced to model changes to a given network that occur throughout a fixed period of time. We introduce and investigate the Temporal Δ Independent Set problem, a temporal variant of the well known Independent Set problem. This problem is e.g. motivated in the context of finding conflict-free schedules for maximum subsets of tasks, that have certain (changing) constraints on each day they need to be performed. We are specifically interested in the case where each task needs to be performed in a certain time-interval on each day and two tasks are in conflict on a day if their time-intervals overlap on that day. This leads us to considering Temporal Δ Independent Set on the restricted class of temporal unit interval graphs, i.e., temporal graphs where each layer is unit interval. We present several hardness results for this problem, as well as two algorithms: The first is a constant-factor approximation algorithm for instances where τ, the total number of time steps (layers) of the temporal graph, and Δ, a parameter that allows us to model some tolerance in the conflicts, are constants. For the second result we use the notion of order preservation for temporal unit interval graphs that, informally, requires the intervals of every layer to obey a common ordering. We provide an FPT algorithm parameterized by the size of minimum vertex deletion set to order preservation.
Cite as

Danny Hermelin, Yuval Itzhaki, Hendrik Molter, and Rolf Niedermeier. Temporal Unit Interval Independent Sets. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hermelin_et_al:LIPIcs.SAND.2022.19,
  author =	{Hermelin, Danny and Itzhaki, Yuval and Molter, Hendrik and Niedermeier, Rolf},
  title =	{{Temporal Unit Interval Independent Sets}},
  booktitle =	{1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-224-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{221},
  editor =	{Aspnes, James and Michail, Othon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.19},
  URN =		{urn:nbn:de:0030-drops-159617},
  doi =		{10.4230/LIPIcs.SAND.2022.19},
  annote =	{Keywords: Temporal Graphs, Vertex Orderings, Order Preservation, Interval Graphs, Algorithms and Complexity}
}
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