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Answering Related Questions

Author Édouard Bonnet

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LIPIcs.ESA.2026.73.pdf
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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
Keywords
  • Sidestep problem
  • hardness radius
  • edge edit distance
  • degree-bounded edge edits
  • robust reductions
  • classical graph problems
  • NP-hardness

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Abstract

We introduce the meta-problem Sidestep(Π, dist, d) for a problem Π, a metric dist over its inputs, and a map d: ℕ → ℝ_+ ∪ {∞}. A solution to Sidestep(Π, dist, d) on an input I of Π is a pair (J, Π(J)) such that dist(I,J) ⩽ d(|I|) and Π(J) is a correct answer to Π on input J. This formalizes the notion of answering a related question (or sidestepping the question), for which we give some motivations, and compare it with the adjacent concepts of smoothed analysis, certified algorithms, planted problems, modification problems, and approximation algorithms. Informally, we call hardness radius the "largest" d such that Sidestep(Π, dist, d) is NP-hard. This framework calls for establishing the hardness radius of problems Π of interest for the relevant distances dist.
We exemplify it with graph problems and two distances dist_∆ and dist_e (the edge edit distance) such that dist_∆(G,H) (resp. dist_e(G,H)) is the maximum degree (resp. number of edges) of the symmetric difference of G and H if these graphs are on the same vertex set, and +∞ otherwise. Thus when solving Sidestep(Π, dist_d, d) (resp. Sidestep(Π, dist_e, d)) on an n-vertex input G, acceptably close graphs H are obtained by XORing G with a graph of maximum degree at most d(n) (resp. having at most d(n) edges). We show that the decision problems Independent Set, Clique, Vertex Cover, Coloring, Clique Cover have hardness radius n^{1/2-o(1)} for dist_∆, and n^{4/3-o(1)} for dist_e, that Hamiltonian Cycle (or Hamiltonian Path) has hardness radius 0 for dist_∆, and somewhere between n^{1/2-o(1)} and n/3 for dist_e, and that Dominating Set has hardness radius n^{1-o(1)} for dist_e. We leave several open questions.

Cite As Get BibTex

Édouard Bonnet. Answering Related Questions. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.73

Author Details

Édouard Bonnet
  • CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, 69342 Lyon, France

Acknowledgements

The author thanks Bart M. P. Jansen for telling him about certified algorithms.

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