Fixed-Parameter Sensitivity Oracles

TL;DR

This paper introduces sensitivity oracles for fixed-parameter tractable graph problems, enabling fast responses to edge failures without recomputing solutions from scratch, with applications to k-Path, k-Vertex Cover, and distance preservation.

Contribution

It combines distance sensitivity oracles with fixed-parameter tractability to create efficient, preprocessed oracles for FPT problems and distance computations under edge failures.

Findings

Designed sensitivity oracles for k-Path and k-Vertex Cover.

Developed a distance-preserving oracle for directed unweighted graphs.

Achieved query times significantly faster than re-running FPT algorithms.

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 $G-F$.