17 Search Results for "Rau, Malin"

Document
DOI: 10.4230/LIPIcs.STACS.2026.54
A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm

Authors: Stefan Hougardy and Bart Zondervan

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
In the Strip Packing problem, we are given a vertical strip of fixed width and unbounded height, along with a set of axis‑parallel rectangles. The task is to place all rectangles within the strip, without overlaps, while minimizing the height of the packing. This problem is known to be NP-hard. The Bottom-Left Algorithm is a simple and widely used heuristic for Strip Packing. Given a fixed order of the rectangles, it places them one by one, always choosing the lowest feasible position in the strip and, in case of ties, the leftmost one. Baker, Coffman, and Rivest proved in 1980 that the Bottom-Left Algorithm has approximation ratio 3 if the rectangles are sorted by decreasing width [Brenda S. Baker et al., 1980]. For the past 45 years, no alternative ordering has been found that improves this bound. We introduce a new rectangle ordering and show that with this ordering the Bottom-Left Algorithm achieves a 13/6 approximation for the Strip Packing problem.
Cite as

Stefan Hougardy and Bart Zondervan. A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hougardy_et_al:LIPIcs.STACS.2026.54,
  author =	{Hougardy, Stefan and Zondervan, Bart},
  title =	{{A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{54:1--54:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle 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.2026.54},
  URN =		{urn:nbn:de:0030-drops-255432},
  doi =		{10.4230/LIPIcs.STACS.2026.54},
  annote =	{Keywords: Approximation Algorithm, Strip Packing, Bottom-Left Algorithm, Rectangle Packing}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.56
A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling

Authors: Klaus Jansen and Felix Ohnesorge

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
In moldable job scheduling, we are provided m identical machines and n jobs that can be executed on a variable number of machines. The execution time of each job depends on the number of machines assigned to execute that job. For the specific problem of monotone moldable job scheduling, jobs are assumed to have a processing time that is non-increasing in the number of machines. The previous best-known algorithms are: (1) a Polynomial Time Approximation Scheme (PTAS) with time complexity Ω(n^{g(1/ε)}), where g(⋅) is a super-exponential function [Jansen and Thöle '08; Jansen and Land '18], (2) a Fully Polynomial Time Approximation Scheme (FPTAS) for the case of m ≥ 8n/(ε) [Jansen and Land '18], and (3) a 3/2 approximation with time complexity O(nmlog(mn)) [Wu, Zhang, and Chen '23]. We present a new practically efficient algorithm with an approximation ratio of ≈ (1.4593 + ε) and a time complexity of O(nm log 1/(ε)). Our result also applies to the contiguous variant of the problem. In addition to our theoretical results, we implement the presented algorithm and show that the practical performance is significantly better than the theoretical worst-case approximation ratio.
Cite as

Klaus Jansen and Felix Ohnesorge. A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jansen_et_al:LIPIcs.STACS.2026.56,
  author =	{Jansen, Klaus and Ohnesorge, Felix},
  title =	{{A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{56:1--56:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle 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.2026.56},
  URN =		{urn:nbn:de:0030-drops-255453},
  doi =		{10.4230/LIPIcs.STACS.2026.56},
  annote =	{Keywords: computing, machine scheduling, moldable, polynomial approximation}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.IPEC.2025.1
A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk)

Authors: Martin Koutecký

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Integer Programming (IP) is a fundamental but computationally hard problem. Still, certain efficiently solvable subclasses have been identified over time, most notably totally unimodular IPs in the 1950s, and fixed-dimension IPs in the 1980s. Starting around the year 2000, a stream of research has identified block-structured IPs as yet another tractable subclass. In this paper, we give a brief and incomplete review of this history, with a focus on several of the author’s contributions.
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Martin Koutecký. A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk). In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{koutecky:LIPIcs.IPEC.2025.1,
  author =	{Kouteck\'{y}, Martin},
  title =	{{A Brief History of Parameterized Algorithms for Block-Structured Integer Programs}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.1},
  URN =		{urn:nbn:de:0030-drops-251338},
  doi =		{10.4230/LIPIcs.IPEC.2025.1},
  annote =	{Keywords: Integer Programming, Parameterized Algorithm, Graver Basis, Treedepth, n-fold, tree-fold, 2-stage stochastic, multistage stochastic, Mixed-Integer Programming}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.15
New Algorithm for Combinatorial n-Folds and Applications

Authors: Klaus Jansen, Kai Kahler, Lis Pirotton, and Malte Tutas

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Block-structured integer linear programs (ILPs) play an important role in various application fields. We address n-fold ILPs where the matrix 𝒜 has a specific structure, i.e., where the blocks in the lower part of 𝒜 consist only of the row vectors (1,… ,1). In this paper, we propose an approach tailored to exactly these combinatorial n-folds. We utilize a divide and conquer approach to separate the original problem such that the right-hand side iteratively decreases in size. We show that this decrease in size can be calculated such that we only need to consider a bounded amount of possible right-hand sides. This, in turn, lets us efficiently combine solutions of the smaller right-hand sides to solve the original problem. We can decide the feasibility of, and also optimally solve, such problems in time (n r Δ)^O(r) log(‖b‖_∞), where n is the number of blocks, r the number of rows in the upper blocks and Δ = ‖A‖_∞. We complement the algorithm by discussing applications of the n-fold ILPs with the specific structure we require. We consider the problems of (i) scheduling on uniform machines, (ii) closest string and (iii) (graph) imbalance. Regarding (i), our algorithm results in running times of p_max^O(d)|I|^O(1), matching a lower bound derived via ETH. For (ii) we achieve running times matching the current state-of-the-art in the general case. In contrast to the state-of-the-art, our result can leverage a bounded number of column-types to yield an improved running time. For (iii), we improve the parameter dependency on the size of the vertex cover.
Cite as

Klaus Jansen, Kai Kahler, Lis Pirotton, and Malte Tutas. New Algorithm for Combinatorial n-Folds and Applications. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2025.15,
  author =	{Jansen, Klaus and Kahler, Kai and Pirotton, Lis and Tutas, Malte},
  title =	{{New Algorithm for Combinatorial n-Folds and Applications}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.15},
  URN =		{urn:nbn:de:0030-drops-251472},
  doi =		{10.4230/LIPIcs.IPEC.2025.15},
  annote =	{Keywords: integer linear programming, n-fold, parameterized complexity, scheduling, uniform machines}
}
Document
DOI: 10.4230/LIPIcs.DISC.2025.27
On the h-Majority Dynamics with Many Opinions

Authors: Francesco d'Amore, Niccolò D'Archivio, George Giakkoupis, and Emanuele Natale

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract
We present the first upper bound on the convergence time to consensus of the well-known h-majority dynamics with k opinions, in the synchronous setting, for h and k that are both non-constant values. We suppose that, at the beginning of the process, there is some initial additive bias towards some plurality opinion, that is, there is an opinion that is supported by x nodes while any other opinion is supported by strictly fewer nodes. We prove that, with high probability, if the bias is ω(√x) and the initial plurality opinion is supported by at least x = ω(log n) nodes, then the process converges to plurality consensus in O(log n) rounds whenever h = ω(n log n / x). A main corollary is the following: if k = o(n / log n) and the process starts from an almost-balanced configuration with an initial bias of magnitude ω(√{n/k}) towards the initial plurality opinion, then any function h = ω(k log n) suffices to guarantee convergence to consensus in O(log n) rounds, with high probability. Our upper bound shows that the lower bound of Ω(k / h²) rounds to reach consensus given by Becchetti et al. (2017) cannot be pushed further than Ω̃(k / h). Moreover, the bias we require is asymptotically smaller than the Ω(√{nlog n}) bias that guarantees plurality consensus in the 3-majority dynamics: in our case, the required bias is at most any (arbitrarily small) function in ω(√x) for any value of k ≥ 2.
Cite as

Francesco d'Amore, Niccolò D'Archivio, George Giakkoupis, and Emanuele Natale. On the h-Majority Dynamics with Many Opinions. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 27:1-27:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{damore_et_al:LIPIcs.DISC.2025.27,
  author =	{d'Amore, Francesco and D'Archivio, Niccol\`{o} and Giakkoupis, George and Natale, Emanuele},
  title =	{{On the h-Majority Dynamics with Many Opinions}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{27:1--27:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.27},
  URN =		{urn:nbn:de:0030-drops-248448},
  doi =		{10.4230/LIPIcs.DISC.2025.27},
  annote =	{Keywords: Distributed Algorithms, Randomized Algorithms, Markov Chains, Consensus Problem, Opinion dynamics, Plurality Consensus}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.10
Improved Approximation Guarantees for Advertisement Placement

Authors: Waldo Gálvez, Roberto Oliva, and Victor Verdugo

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
The advertisement placement problem involves selecting and scheduling ads within a timeline that has capacity constraints to maximize profit. Each task is characterized by its height, width, and profit, and must be fully scheduled across multiple time slots. This problem models practical scenarios such as internet advertising and energy management, and it also generalizes classical combinatorial optimization problems like the knapsack and bin packing problems. We present a simple (2+ε)-approximation algorithm for any ε > 0, which improves upon the state-of-the-art 3+ε factor established by Freund and Naor twenty years ago. Our approach combines rounding techniques with dynamic programming and an efficient extension of list scheduling. Furthermore, we enhance this method with linear programming techniques to provide an almost optimal (1+ε)-approximation algorithm under resource augmentation, which allows for a slight increase in time slot capacities.
Cite as

Waldo Gálvez, Roberto Oliva, and Victor Verdugo. Improved Approximation Guarantees for Advertisement Placement. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{galvez_et_al:LIPIcs.APPROX/RANDOM.2025.10,
  author =	{G\'{a}lvez, Waldo and Oliva, Roberto and Verdugo, Victor},
  title =	{{Improved Approximation Guarantees for Advertisement Placement}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.10},
  URN =		{urn:nbn:de:0030-drops-243762},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.10},
  annote =	{Keywords: Advertisement Placement, Two-dimensional Packing, Geometric Knapsack, Resource Allocation}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.104
Improved Approximation Algorithms for Three-Dimensional Bin Packing

Authors: Debajyoti Kar, Arindam Khan, and Malin Rau

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study three fundamental three-dimensional (3D) geometric packing problems: 3D (Geometric) Bin Packing (3D-BP), 3D Strip Packing (3D-SP), and Minimum Volume Bounding Box (3D-MVBB), where given a set of 3D (rectangular) cuboids, the goal is to find an axis-aligned nonoverlapping packing of all cuboids. In 3D-BP, we need to pack the given cuboids into the minimum number of unit cube bins. In 3D-SP, we need to pack them into a 3D cuboid with a unit square base and minimum height. Finally, in 3D-MVBB, the goal is to pack into a cuboid box of minimum volume. It is NP-hard to even decide whether a set of rectangles can be packed into a unit square bin - giving an (absolute) approximation hardness of 2 for 3D-BP and 3D-SP. The previous best (absolute) approximation for all three problems is by Li and Cheng (SICOMP, 1990), who gave algorithms with approximation ratios of 13, 46/7, and 46/7+ε, respectively, for 3D-BP, 3D-SP, and 3D-MVBB. We provide improved approximation ratios of 6, 6, and 3+ε, respectively, for the three problems, for any constant ε > 0. For 3D-BP, in the asymptotic regime, Bansal, Correa, Kenyon, and Sviridenko (Math. Oper. Res., 2006) showed that there is no asymptotic polynomial-time approximation scheme (APTAS) even when all items have the same height. Caprara (Math. Oper. Res., 2008) gave an asymptotic approximation ratio of T_{∞}² + ε ≈ 2.86, where T_{∞} is the well-known Harmonic constant in Bin Packing. We provide an algorithm with an improved asymptotic approximation ratio of 3 T_{∞}/2 + ε ≈ 2.54. Further, we show that unlike 3D-BP (and 3D-SP), 3D-MVBB admits an APTAS.
Cite as

Debajyoti Kar, Arindam Khan, and Malin Rau. Improved Approximation Algorithms for Three-Dimensional Bin Packing. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 104:1-104:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kar_et_al:LIPIcs.ICALP.2025.104,
  author =	{Kar, Debajyoti and Khan, Arindam and Rau, Malin},
  title =	{{Improved Approximation Algorithms for Three-Dimensional Bin Packing}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{104:1--104:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.104},
  URN =		{urn:nbn:de:0030-drops-234814},
  doi =		{10.4230/LIPIcs.ICALP.2025.104},
  annote =	{Keywords: Approximation Algorithms, Geometric Packing, Multidimensional Packing}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.112
Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns

Authors: Alexandra Lassota and Koen Ligthart

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study integer linear programs (ILP) of the form min{c^⊤ x | Ax = b,l ≤ x ≤ u,x ∈ ℤⁿ} and analyze their parameterized complexity with respect to their distance to the generalized matching problem, following the well-established approach of capturing the hardness of a problem by the distance to triviality. The generalized matching problem is an ILP where each column of the constraint matrix has 1-norm of at most 2. It captures several well-known polynomial time solvable problems such as matching and flow problems. We parameterize by the size of variable and constraint backdoors, which measure the least number of columns or rows that must be deleted to obtain a generalized matching ILP. This extends generalized matching problems by allowing a parameterized number of additional arbitrary variables or constraints, yielding a novel parameter. We present the following results: (i) a fixed-parameter tractable (FPT) algorithm for ILPs parameterized by the size p of a minimum variable backdoor to generalized matching; (ii) a randomized slice-wise polynomial (XP) time algorithm for ILPs parameterized by the size h of a minimum constraint backdoor to generalized matching as long as c and A are encoded in unary; (iii) we complement (ii) by proving that solving an ILP is W[1]-hard when parameterized by h even when c,A,l,u have coefficients of constant size. To obtain (i), we prove a variant of lattice-convexity of the degree sequences of weighted b-matchings, which we study in the light of SBO jump M-convex functions. This allows us to model the matching part as a polyhedral constraint on the integer backdoor variables. The resulting ILP is solved in FPT time using an integer programming algorithm. For (ii), the randomized XP time algorithm is obtained by pseudo-polynomially reducing the problem to the exact matching problem. To prevent an exponential blowup in terms of the encoding length of b, we bound the Graver complexity of the constraint matrix and employ a Graver augmentation local search framework. The hardness result (iii) is obtained through a parameterized reduction from ILP with h constraints and coefficients encoded in unary.
Cite as

Alexandra Lassota and Koen Ligthart. Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 112:1-112:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lassota_et_al:LIPIcs.ICALP.2025.112,
  author =	{Lassota, Alexandra and Ligthart, Koen},
  title =	{{Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{112:1--112:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.112},
  URN =		{urn:nbn:de:0030-drops-234895},
  doi =		{10.4230/LIPIcs.ICALP.2025.112},
  annote =	{Keywords: Integer Programming, fixed-parameter Tractability, polyhedral Optimization, Matchings}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.18
Dynamic Averaging Load Balancing on Arbitrary Graphs

Authors: Petra Berenbrink, Lukas Hintze, Hamed Hosseinpour, Dominik Kaaser, and Malin Rau

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and deterministic sequences of matchings covering the whole graph. We bound the discrepancy, which is defined as the difference between the maximum and the minimum load. Our results cover a broad range of graph classes and, to the best of our knowledge, our analysis is the first result for discrete and dynamic averaging load balancing processes. As our main technical contribution we develop a drift result that allows us to apply techniques based on the effective resistance in an electrical network to the setting of dynamic load balancing.
Cite as

Petra Berenbrink, Lukas Hintze, Hamed Hosseinpour, Dominik Kaaser, and Malin Rau. Dynamic Averaging Load Balancing on Arbitrary Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.ICALP.2023.18,
  author =	{Berenbrink, Petra and Hintze, Lukas and Hosseinpour, Hamed and Kaaser, Dominik and Rau, Malin},
  title =	{{Dynamic Averaging Load Balancing on Arbitrary Graphs}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{18:1--18:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.18},
  URN =		{urn:nbn:de:0030-drops-180707},
  doi =		{10.4230/LIPIcs.ICALP.2023.18},
  annote =	{Keywords: Dynamic Load Balancing, Distributed Computing, Randomized Algorithms, Drift Analysis}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2022.23
On the Hierarchy of Distributed Majority Protocols

Authors: Petra Berenbrink, Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Dominik Kaaser, and Malin Rau

Published in: LIPIcs, Volume 253, 26th International Conference on Principles of Distributed Systems (OPODIS 2022)

Abstract
We study the consensus problem among n agents, defined as follows. Initially, each agent holds one of two possible opinions. The goal is to reach a consensus configuration in which every agent shares the same opinion. To this end, agents randomly sample other agents and update their opinion according to a simple update function depending on the sampled opinions. We consider two communication models: the gossip model and a variant of the population model. In the gossip model, agents are activated in parallel, synchronous rounds. In the population model, one agent is activated after the other in a sequence of discrete time steps. For both models we analyze the following natural family of majority processes called j-Majority: when activated, every agent samples j other agents uniformly at random (with replacement) and adopts the majority opinion among the sample (breaking ties uniformly at random). As our main result we show a hierarchy among majority protocols: (j+1)-Majority (for j > 1) converges stochastically faster than j-Majority for any initial opinion configuration. In our analysis we use Strassen’s Theorem to prove the existence of a coupling. This gives an affirmative answer for the case of two opinions to an open question asked by Berenbrink et al. [PODC 2017].
Cite as

Petra Berenbrink, Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Dominik Kaaser, and Malin Rau. On the Hierarchy of Distributed Majority Protocols. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.OPODIS.2022.23,
  author =	{Berenbrink, Petra and Coja-Oghlan, Amin and Gebhard, Oliver and Hahn-Klimroth, Max and Kaaser, Dominik and Rau, Malin},
  title =	{{On the Hierarchy of Distributed Majority Protocols}},
  booktitle =	{26th International Conference on Principles of Distributed Systems (OPODIS 2022)},
  pages =	{23:1--23:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-265-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{253},
  editor =	{Hillel, Eshcar and Palmieri, Roberto and Rivi\`{e}re, Etienne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2022.23},
  URN =		{urn:nbn:de:0030-drops-176434},
  doi =		{10.4230/LIPIcs.OPODIS.2022.23},
  annote =	{Keywords: Consensus, Majority, Hierarchy, Stochastic Dominance, Population Protocols, Gossip Model, Strassen’s Theorem}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.21
Peak Demand Minimization via Sliced Strip Packing

Authors: Max A. Deppert, Klaus Jansen, Arindam Khan, Malin Rau, and Malte Tutas

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We study the Nonpreemptive Peak Demand Minimization (NPDM) problem, where we are given a set of jobs, specified by their processing times and energy requirements. The goal is to schedule all jobs within a fixed time period such that the peak load (the maximum total energy requirement at any time) is minimized. This problem has recently received significant attention due to its relevance in smart-grids. Theoretically, the problem is related to the classical strip packing problem (SP). In SP, a given set of axis-aligned rectangles must be packed into a fixed-width strip, such that the height of the strip is minimized. NPDM can be modeled as strip packing with slicing and stacking constraint: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions where two slices of the same rectangle are intersected by the same vertical line. Nonpreemption enforces the slices to be placed in contiguous horizontal locations (but may be placed at different vertical locations). We obtain a (5/3+ε)-approximation algorithm for the problem. We also provide an asymptotic efficient polynomial-time approximation scheme (AEPTAS) which generates a schedule for almost all jobs with energy consumption (1+ε) OPT. The remaining jobs fit into a thin container of height 1. The previous best result for NPDM was a 2.7 approximation based on FFDH [Ranjan et al., 2015]. One of our key ideas is providing several new lower bounds on the optimal solution of a geometric packing, which could be useful in other related problems. These lower bounds help us to obtain approximative solutions based on Steinberg’s algorithm in many cases. In addition, we show how to split schedules generated by the AEPTAS into few segments and to rearrange the corresponding jobs to insert the thin container mentioned above.
Cite as

Max A. Deppert, Klaus Jansen, Arindam Khan, Malin Rau, and Malte Tutas. Peak Demand Minimization via Sliced Strip Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{deppert_et_al:LIPIcs.APPROX/RANDOM.2021.21,
  author =	{Deppert, Max A. and Jansen, Klaus and Khan, Arindam and Rau, Malin and Tutas, Malte},
  title =	{{Peak Demand Minimization via Sliced Strip Packing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{21:1--21:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.21},
  URN =		{urn:nbn:de:0030-drops-147145},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.21},
  annote =	{Keywords: scheduling, peak demand minimization, approximation}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.53
Closing the Gap for Single Resource Constraint Scheduling

Authors: Klaus Jansen and Malin Rau

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
In the problem called single resource constraint scheduling, we are given m identical machines and a set of jobs, each needing one machine to be processed as well as a share of a limited renewable resource R. A schedule of these jobs is feasible if, at each point in the schedule, the number of machines and resources required by jobs processed at this time is not exceeded. It is NP-hard to approximate this problem with a ratio better than 3/2. On the other hand, the best algorithm so far has an absolute approximation ratio of 2+ε. In this paper, we present an algorithm with absolute approximation ratio (3/2+ε), which closes the gap between inapproximability and best algorithm with exception of a negligible small ε.
Cite as

Klaus Jansen and Malin Rau. Closing the Gap for Single Resource Constraint Scheduling. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jansen_et_al:LIPIcs.ESA.2021.53,
  author =	{Jansen, Klaus and Rau, Malin},
  title =	{{Closing the Gap for Single Resource Constraint Scheduling}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{53:1--53:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.53},
  URN =		{urn:nbn:de:0030-drops-146344},
  doi =		{10.4230/LIPIcs.ESA.2021.53},
  annote =	{Keywords: resource constraint scheduling, approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.11
Solving Packing Problems with Few Small Items Using Rainbow Matchings

Authors: Max Bannach, Sebastian Berndt, Marten Maack, Matthias Mnich, Alexandra Lassota, Malin Rau, and Malte Skambath

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number k of small items. Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by k. The algorithms are randomized with one-sided error and run in time 4^k⋅ k!⋅ n^{O(1)}. To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. We also present a deterministic fixed-parameter algorithm for Bin Covering with run time O((k!)² ⋅ k ⋅ 2^k ⋅ n log(n)).
Cite as

Max Bannach, Sebastian Berndt, Marten Maack, Matthias Mnich, Alexandra Lassota, Malin Rau, and Malte Skambath. Solving Packing Problems with Few Small Items Using Rainbow Matchings. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bannach_et_al:LIPIcs.MFCS.2020.11,
  author =	{Bannach, Max and Berndt, Sebastian and Maack, Marten and Mnich, Matthias and Lassota, Alexandra and Rau, Malin and Skambath, Malte},
  title =	{{Solving Packing Problems with Few Small Items Using Rainbow Matchings}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.11},
  URN =		{urn:nbn:de:0030-drops-126816},
  doi =		{10.4230/LIPIcs.MFCS.2020.11},
  annote =	{Keywords: Bin Packing, Knapsack, matching, fixed-parameter tractable}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.44
A Tight (3/2+ε) Approximation for Skewed Strip Packing

Authors: Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, Klaus Jansen, Arindam Khan, and Malin Rau

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
In the Strip Packing problem, we are given a vertical half-strip [0,W]× [0,+∞) and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. Strip Packing generalizes classical well-studied problems such as Makespan Minimization on identical machines (when rectangle widths are identical) and Bin Packing (when rectangle heights are identical). It has applications in manufacturing, scheduling and energy consumption in smart grids among others. It is NP-hard to approximate this problem within a factor (3/2-ε) for any constant ε > 0 by a simple reduction from the Partition problem. The current best approximation factor for Strip Packing is (5/3+ε) by Harren et al. [Computational Geometry '14], and it is achieved with a fairly complex algorithm and analysis. It seems plausible that Strip Packing admits a (3/2+ε)-approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter δ > 0, we call large the rectangles with width at least δ W and height at least δ OPT, and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, it is NP-hard to approximate within a factor (3/2-ε) and we provide an (almost) tight (3/2+ε)-approximation algorithm.
Cite as

Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, Klaus Jansen, Arindam Khan, and Malin Rau. A Tight (3/2+ε) Approximation for Skewed Strip Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{galvez_et_al:LIPIcs.APPROX/RANDOM.2020.44,
  author =	{G\'{a}lvez, Waldo and Grandoni, Fabrizio and Ameli, Afrouz Jabal and Jansen, Klaus and Khan, Arindam and Rau, Malin},
  title =	{{A Tight (3/2+\epsilon) Approximation for Skewed Strip Packing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{44:1--44:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.44},
  URN =		{urn:nbn:de:0030-drops-126478},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.44},
  annote =	{Keywords: strip packing, approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.5
Inapproximability Results for Scheduling with Interval and Resource Restrictions

Authors: Marten Maack and Klaus Jansen

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions. For the case of interval restrictions - where the machines can be totally ordered such that jobs are eligible on consecutive machines - we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known. Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 ≈ 1.02 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives.
Cite as

Marten Maack and Klaus Jansen. Inapproximability Results for Scheduling with Interval and Resource Restrictions. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{maack_et_al:LIPIcs.STACS.2020.5,
  author =	{Maack, Marten and Jansen, Klaus},
  title =	{{Inapproximability Results for Scheduling with Interval and Resource Restrictions}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.5},
  URN =		{urn:nbn:de:0030-drops-118663},
  doi =		{10.4230/LIPIcs.STACS.2020.5},
  annote =	{Keywords: Scheduling, Restricted Assignment, Approximation, Inapproximability, PTAS}
}
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