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.
|Title of host publication||47th International Colloquium on Automata, Languages, and Programming, ICALP 2020|
|Editors||Artur Czumaj, Anuj Dawar, Emanuela Merelli|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Jun 2020|
|Event||47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 - Virtual, Online, Germany|
Duration: 8 Jul 2020 → 11 Jul 2020
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||47th International Colloquium on Automata, Languages, and Programming, ICALP 2020|
|Period||8/07/20 → 11/07/20|
Bibliographical noteFunding Information:
Funding Shon Feller: Supported in part by Israel Science Foundation grant 592/17. Oren Weimann: Supported in part by Israel Science Foundation grant 592/17.
© Amir Abboud, Shon Feller, and Oren Weimann; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).
- All-pairs shortest paths
- Distance product
- Fine-grained complexity
- Min-plus convolution
- Parity problems
ASJC Scopus subject areas