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Documents authored by Nielsen, Mads Anker

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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.142
Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules

Authors: Mirabel Mendoza-Cadena, Arturo Merino, Mads Anker Nielsen, and Kevin Schewior

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
This paper considers a framework for combinatorial variants of perpetual-scheduling problems. Given an independence system (E,ℐ), a schedule consists of an independent set I_t ∈ ℐ for every time step t ∈ ℕ, with the objective of fulfilling frequency requirements on the occurrence of elements in E. We focus specifically on combinatorial bamboo garden trimming, where elements accumulate height at growth rates g(e) for e ∈ E and are reset to zero when scheduled, with the goal of minimizing the maximum height attained by any element. We assume that g is normalized so that it is a convex combination of the incidence vectors of ℐ. Using the integrality of the matroid-intersection polytope, we prove that, when (E,ℐ) is a matroid, it is possible to guarantee a maximum height of at most 2, which is optimal. We complement this existential result with efficient algorithms for specific matroid classes, achieving a maximum height of 2 for uniform and partition matroids, and 4 for graphic and laminar matroids. In contrast, we show that for general independence systems, the optimal guaranteed height is Θ(log |E|) and can be achieved by an efficient algorithm. For combinatorial pinwheel scheduling, where each element e ∈ E needs to occur in the schedule at least every a_e ∈ ℕ time steps, our results imply bounds on the density sufficient for schedulability.
Cite as

Mirabel Mendoza-Cadena, Arturo Merino, Mads Anker Nielsen, and Kevin Schewior. Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 142:1-142:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{mendozacadena_et_al:LIPIcs.ICALP.2026.142,
  author =	{Mendoza-Cadena, Mirabel and Merino, Arturo and Nielsen, Mads Anker and Schewior, Kevin},
  title =	{{Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{142:1--142:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael 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.2026.142},
  URN =		{urn:nbn:de:0030-drops-265319},
  doi =		{10.4230/LIPIcs.ICALP.2026.142},
  annote =	{Keywords: Perpetual Scheduling, Matroids, Bamboo Garden Trimming, Pinwheel}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.26
Non-Adaptive Evaluation of k-of- n Functions: Tight Gap and a Unit-Cost PTAS

Authors: Mads Anker Nielsen, Lars Rohwedder, and Kevin Schewior

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

Abstract
We consider the Stochastic Boolean Function Evaluation (SBFE) problem in the well-studied case of k-of-n functions: There are independent Boolean random variables x_1,… ,x_n where each variable i has a known probability p_i of taking value 1, and a known cost c_i that can be paid to find out its value. The value of the function is 1 iff there are at least k 1s among the variables. The goal is to efficiently compute a strategy that, at minimum expected cost, tests the variables until the function value is determined. While an elegant polynomial-time exact algorithm is known when tests can be made adaptively, we focus on the non-adaptive variant, for which much less is known. First, we show a clean and tight lower bound of 2 on the adaptivity gap, i.e., the worst-case multiplicative loss in the objective function caused by disallowing adaptivity, of the problem. This improves the tight lower bound of 3/2 for the unit-cost variant. Second, we give a PTAS for computing the best non-adaptive strategy in the unit-cost case, the first PTAS for an SBFE problem. At the core, our scheme establishes a novel notion of two-sided dominance (w.r.t. the optimal solution) by guessing so-called milestone tests for a set of carefully chosen buckets of tests. To turn this technique into a polynomial-time algorithm, we use a decomposition approach paired with a random-shift argument.
Cite as

Mads Anker Nielsen, Lars Rohwedder, and Kevin Schewior. Non-Adaptive Evaluation of k-of- n Functions: Tight Gap and a Unit-Cost PTAS. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{nielsen_et_al:LIPIcs.APPROX/RANDOM.2025.26,
  author =	{Nielsen, Mads Anker and Rohwedder, Lars and Schewior, Kevin},
  title =	{{Non-Adaptive Evaluation of k-of- n Functions: Tight Gap and a Unit-Cost PTAS}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{26:1--26:18},
  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.2025.26},
  URN =		{urn:nbn:de:0030-drops-243920},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.26},
  annote =	{Keywords: Approximation scheme, Boolean functions, stochastic combinatorial optimization, stochastic function evaluation, sequential testing, adaptivity}
}
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