eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-03-03
35:1
35:19
10.4230/LIPIcs.STACS.2023.35
article
Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions
Heeger, Klaus
1
https://orcid.org/0000-0001-8779-0890
Nichterlein, André
1
https://orcid.org/0000-0001-7451-9401
Niedermeier, Rolf
1
https://orcid.org/0000-0003-1703-1236
Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
We provide a general framework to exclude parameterized running times of the form O(l^β + n^γ) for problems that have polynomial running time lower bounds under hypotheses from fine-grained complexity. Our framework is based on cross-compositions from parameterized complexity. We (conditionally) exclude running times of the form O(l^{γ/(γ-1) - ε} + n^γ) for any 1 < γ < 2 and ε > 0 for the following problems:
- Longest Common (Increasing) Subsequence: Given two length-n strings over an alphabet Σ (over ℕ) and l ∈ ℕ, is there a common (increasing) subsequence of length l in both strings?
- Discrete Fréchet Distance: Given two lists of n points each and k ∈ N, is the Fréchet distance of the lists at most k? Here l is the maximum number of points which one list is ahead of the other list in an optimum traversal.
- Planar Motion Planning: Given a set of n non-intersecting axis-parallel line segment obstacles in the plane and a line segment robot (called rod), can the rod be moved to a specified target without touching any obstacles? Here l is the maximum number of segments any segment has in its vicinity. Moreover, we exclude running times O(l^{2γ/(γ-1) - ε} + n^γ) for any 1 < γ < 3 and ε > 0 for:
- Negative Triangle: Given an edge-weighted graph with n vertices, is there a triangle whose sum of edge-weights is negative? Here l is the order of a maximum connected component.
- Triangle Collection: Given a vertex-colored graph with n vertices, is there for each triple of colors a triangle whose vertices have these three colors? Here l is the order of a maximum connected component.
- 2nd Shortest Path: Given an n-vertex edge-weighted digraph, vertices s and t, and k ∈ ℕ, has the second longest s-t-path length at most k? Here l is the directed feedback vertex set number. Except for 2nd Shortest Path all these running time bounds are tight, that is, algorithms with running time O(l^{γ/(γ-1)} + n^γ) for any 1 < γ < 2 and O(l^{2γ/(γ -1)} + n^γ) for any 1 < γ < 3, respectively, are known. Our running time lower bounds also imply lower bounds on kernelization algorithms for these problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol254-stacs2023/LIPIcs.STACS.2023.35/LIPIcs.STACS.2023.35.pdf
FPT in P
Kernelization
Decomposition