Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions

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(^{ℓ2γ/(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(^{ℓγ/(γ−1)} + n^γ) for any 1 < γ < 2 and O(^{ℓ2γ/(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.