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Fixed-Parameter Sensitivity Oracles

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



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Author Details

Davide Bilò
  • Department of Humanities and Social Sciences, University of Sassari, Italy
Katrin Casel
  • Hasso Plattner Institute, University of Potsdam, Germany
Keerti Choudhary
  • Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India
Sarel Cohen
  • School of Computer Science, Tel-Aviv-Yaffo Academic College, Israel
Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
J.A. Gregor Lagodzinski
  • Hasso Plattner Institute, University of Potsdam, Germany
Martin Schirneck
  • Hasso Plattner Institute, University of Potsdam, Germany
Simon Wietheger
  • Hasso Plattner Institute, University of Potsdam, Germany

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Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 23:1-23:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.23

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 Π on a graph G with parameter k preprocesses G in time O(g(f,k) ⋅ poly(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 design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk⋅log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}⋅log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds. Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k²+k}) and a query time of O(f+k²); finally, the third one takes O(fk+k²) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ⋅ n) with query time in O(poly(f,k)), where k is the size of a vertex cover. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we 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. The oracle size is O(2^k k²⋅n), while the time needed to answer a query is O(2^k f^ω k^ω), where ω < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ⋅ n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the Õ(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017].

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
Keywords
  • data structures
  • distance preservers
  • distance sensitivity oracles
  • fault tolerance
  • fixed-parameter tractability
  • k-path
  • vertex cover

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