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Fixed-Parameter Sensitivity Oracles

by   Davide Bilò, et al.
Hasso Plattner Institute
Indian Institute of Technology Delhi
The Academic College of Tel Aviv Yaffo

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 Π on a graph G with parameter k preprocesses G in time O(g(f,k) ·(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the Π-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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