Fixed-Parameter Sensitivity Oracles

Davide Bilò; Katrin Casel; Keerti Choudhary; Sarel Cohen; Tobias Friedrich; J. A. Gregor Lagodzinski; Martin Schirneck; Simon Wietheger

arXiv · 2112.03059

Fixed-Parameter Sensitivity Oracles

Davide Bilò·Katrin Casel·Keerti Choudhary·Sarel Cohen·Tobias Friedrich·J. A. Gregor Lagodzinski·Martin Schirneck·Simon Wietheger

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Abstract

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity $f$ for an FPT problem $\Pi$ on a graph $G$ with parameter $k$ preprocesses $G$ in time $O(g(f,k) \cdot \textsf{poly}(n))$. When queried with a set $F$ of at most $f$ edges of $G$, the oracle reports the answer to the $\Pi$-with the same parameter $k$-on the graph $G-F$, i.e., $G$ deprived of $F$. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on $G-F$ from scratch. We mainly design sensitivity oracles for the $k$-Path and the $k$-Vertex Cover problem. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we also introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source $s$ and parameters $f$ and $k$, to construct a polynomial-sized oracle that efficiently reports, for any target vertex $v$ and set $F$ of at most $f$ edges, whether the distance from $s$ to $v$ increases at most by an additive term of $k$ in $G-F$.

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BibTeX RIS

Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J. A. Gregor Lagodzinski, Martin Schirneck, Simon Wietheger. 2021-12-06. Fixed-Parameter Sensitivity Oracles. https://arxiv.org/abs/2112.03059

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