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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2020.5

On the Fine-Grained Complexity of Parity Problems

Authors: Amir Abboud, Shon Feller, and Oren Weimann

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract

We consider the parity variants of basic problems studied in fine-grained complexity. We show that finding the exact solution is just as hard as finding its parity (i.e. if the solution is even or odd) for a large number of classical problems, including All-Pairs Shortest Paths (APSP), Diameter, Radius, Median, Second Shortest Path, Maximum Consecutive Subsums, Min-Plus Convolution, and 0/1-Knapsack. A direct reduction from a problem to its parity version is often difficult to design. Instead, we revisit the existing hardness reductions and tailor them in a problem-specific way to the parity version. Nearly all reductions from APSP in the literature proceed via the (subcubic-equivalent but simpler) Negative Weight Triangle (NWT) problem. Our new modified reductions also start from NWT or a non-standard parity variant of it. We are not able to establish a subcubic-equivalence with the more natural parity counting variant of NWT, where we ask if the number of negative triangles is even or odd. Perhaps surprisingly, we justify this by designing a reduction from the seemingly-harder Zero Weight Triangle problem, showing that parity is (conditionally) strictly harder than decision for NWT.

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Amir Abboud, Shon Feller, and Oren Weimann. On the Fine-Grained Complexity of Parity Problems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{abboud_et_al:LIPIcs.ICALP.2020.5,
  author =	{Abboud, Amir and Feller, Shon and Weimann, Oren},
  title =	{{On the Fine-Grained Complexity of Parity Problems}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{5:1--5:19},
  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.2020.5},
  URN =		{urn:nbn:de:0030-drops-124127},
  doi =		{10.4230/LIPIcs.ICALP.2020.5},
  annote =	{Keywords: All-pairs shortest paths, Fine-grained complexity, Diameter, Distance product, Min-plus convolution, Parity problems}
}

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On the Fine-Grained Complexity of Parity Problems

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