Space-Efficient Graph Kernelizations

TL;DR

The paper addresses space-efficient kernelizations for graph problems parameterized by $k$ and shows kernels whose size is polynomial in $k$ and computable in polynomial time with $O(poly(k) \, log\, n)$ bits, covering Feedback Vertex Set, Path Contraction, and Cluster Editing/Deletion. It introduces a separator-based, subgraph-condensing framework that preserves a poly(k)-sized kernel and operates within the restricted space model. For Path Contraction, it replaces bridge-based reductions with a BFS-separator strategy that bounds the separator by $O(k)$ and shrinks long degree-2 chains, yielding a kernel of $O(k^2)$ vertices/edges and refining to $3k+4$ vertices in $O(n \log k + poly(k))$ time using $O(poly(k) \log n)$ bits. For FVS, the authors develop rules (Loop, Leaf, Chain, Flower) to obtain a full kernel of size $n' = 2k^2 + k$ with time $O(n^5 poly(k))$ and space $O(k^4 \log n)$ bits, and discuss comparisons to prior space-efficient and standard algorithms, including concurrent results by Biswas et al. Overall, kernel cascades yield improved space-bounded kernels in polynomial time.

Abstract

Let $n$ be the size of a parameterized problem and $k$ the parameter. We present kernels for Feedback Vertex Set, Path Contraction and Cluster Editing/Deletion whose sizes are all polynomial in $k$ and that are computable in polynomial time and with $O(\rm{poly}(k) \log n)$ bits (of working memory). By using kernel cascades, we obtain the best known kernels in polynomial time with $O(\rm{poly}(k) \log n)$ bits.