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Documents authored by Dürr, Anita

Document
DOI: 10.4230/LIPIcs.ESA.2024.33
Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing

Authors: Karl Bringmann, Anita Dürr, and Adam Polak

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time Õ(n + t√{p_{max}}), where n is the number of items, t is the knapsack capacity, and p_{max} is the maximum item profit. This improves over the Õ(n + t p_{max})-time algorithm based on the convolution and prediction technique by Bateni et al. (STOC 2018). Moreover, we give some evidence, based on a strengthening of the Min-Plus Convolution Hypothesis, that our running time might be optimal. Our algorithm uses two new technical tools, which might be of independent interest. First, we generalize the Õ(n^{1.5})-time algorithm for bounded monotone min-plus convolution by Chi et al. (STOC 2022) to the rectangular case where the range of entries can be different from the sequence length. Second, we give a reduction from general knapsack instances to balanced instances, where all items have nearly the same profit-to-weight ratio, up to a constant factor. Using these techniques, we can also obtain algorithms that run in time Õ(n + OPT√{w_{max}}), Õ(n + (nw_{max}p_{max})^{1/3}t^{2/3}), and Õ(n + (nw_{max}p_{max})^{1/3} OPT^{2/3}), where OPT is the optimal total profit and w_{max} is the maximum item weight.
Cite as

Karl Bringmann, Anita Dürr, and Adam Polak. Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bringmann_et_al:LIPIcs.ESA.2024.33,
  author =	{Bringmann, Karl and D\"{u}rr, Anita and Polak, Adam},
  title =	{{Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{33:1--33:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.33},
  URN =		{urn:nbn:de:0030-drops-211047},
  doi =		{10.4230/LIPIcs.ESA.2024.33},
  annote =	{Keywords: 0-1-Knapsack problem, bounded monotone min-plus convolution, fine-grained complexity}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.18
An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs

Authors: Anita Dürr, Nicolas El Maalouly, and Lasse Wulf

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph G and an integer k one has to decide whether there exists a perfect matching in G with exactly k red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis. In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly k red edges, not a lot of work focuses on computing perfect matchings with almost k red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS'23). It outputs a perfect matching with k' red edges with the guarantee that 0.5k ≤ k' ≤ 1.5k. In the present paper we aim at approximating the number of red edges without exceeding the limit of k red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with k' red edges such that k/3 ≤ k' ≤ k.
Cite as

Anita Dürr, Nicolas El Maalouly, and Lasse Wulf. An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{durr_et_al:LIPIcs.APPROX/RANDOM.2023.18,
  author =	{D\"{u}rr, Anita and El Maalouly, Nicolas and Wulf, Lasse},
  title =	{{An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{18:1--18:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.18},
  URN =		{urn:nbn:de:0030-drops-188436},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.18},
  annote =	{Keywords: Perfect Matching, Exact Matching, Red-Blue Matching, Approximation Algorithms, Bounded Color Matching}
}
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