On the Fine-Grained Complexity of Parity Problems

Authors Amir Abboud, Shon Feller, Oren Weimann

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Amir Abboud
  • IBM Almaden Research Center, San Jose, CA, USA
Shon Feller
  • University of Haifa, Israel
Oren Weimann
  • University of Haifa, Israel

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • All-pairs shortest paths
  • Fine-grained complexity
  • Diameter
  • Distance product
  • Min-plus convolution
  • Parity problems


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