eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-29
5:1
5:19
10.4230/LIPIcs.ICALP.2020.5
article
On the Fine-Grained Complexity of Parity Problems
Abboud, Amir
1
Feller, Shon
2
Weimann, Oren
2
https://orcid.org/0000-0002-4510-7552
IBM Almaden Research Center, San Jose, CA, USA
University of Haifa, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol168-icalp2020/LIPIcs.ICALP.2020.5/LIPIcs.ICALP.2020.5.pdf
All-pairs shortest paths
Fine-grained complexity
Diameter
Distance product
Min-plus convolution
Parity problems