Kernelization Lower Bounds for Finding Constant-Size Subgraphs

TL;DR

This paper investigates the fundamental limits of kernelization for polynomial-time solvable problems, demonstrating that certain small kernels would contradict well-known complexity conjectures.

Contribution

It provides the first conceptual lower bounds on kernel sizes for specific problems, linking kernelization limits to major complexity conjectures.

Findings

Linear-time strict kernel for negative triangle detection would violate the APSP-conjecture.

Establishes lower bounds on kernel sizes for problems parameterized by graph degree.

Highlights the limitations of kernelization in polynomial-time solvable problems.

Abstract

Kernelization is an important tool in parameterized algorithmics. Given an input instance accompanied by a parameter, the goal is to compute in polynomial time an equivalent instance of the same problem such that the size of the reduced instance only depends on the parameter and not on the size of the original instance. In this paper, we provide a first conceptual study on limits of kernelization for several polynomial-time solvable problems. For instance, we consider the problem of finding a triangle with negative sum of edge weights parameterized by the maximum degree of the input graph. We prove that a linear-time computable strict kernel of truly subcubic size for this problem violates the popular APSP-conjecture.