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.
Original language | English |
---|---|
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 |
ISBN (Electronic) | 9783959771382 |
DOIs | |
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 |
Publication series
Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|
Volume | 168 |
ISSN (Print) | 1868-8969 |
Conference
Conference | 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 |
---|---|
Country/Territory | Germany |
City | Virtual, Online |
Period | 8/07/20 → 11/07/20 |
Bibliographical note
Publisher Copyright:© Amir Abboud, Shon Feller, and Oren Weimann; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).
Keywords
- All-pairs shortest paths
- Diameter
- Distance product
- Fine-grained complexity
- Min-plus convolution
- Parity problems
ASJC Scopus subject areas
- Software