Answering Related Questions
Édouard Bonnet
TL;DR
The paper introduces Sidestep$, a meta-problem that asks for a nearby input and its correct solution, to formalize answering related questions under a tunable distance. It defines two natural graph distances, dist_Δ and dist_e, and analyzes the hardness radius of core graph problems, deriving tight and near-tight bounds for several tasks, including Independent Set, Clique, Vertex Cover, Coloring, Hamiltonian Cycle, and Dominating Set. The results combine constructive sidestep algorithms, robust reductions, and isomorphism-based transfers to extend radii across related problems, yielding a refined landscape of fine-grained hardness under input perturbations. The work also highlights open questions and potential extensions, providing a framework for dataset design and benchmarking in the context of related-question answering on graphs.
Abstract
We introduce the meta-problem Sidestep$(Π, \mathsf{dist}, d)$ for a problem $Π$, a metric $\mathsf{dist}$ over its inputs, and a map $d: \mathbb N \to \mathbb R_+ \cup \{\infty\}$. A solution to Sidestep$(Π, \mathsf{dist}, d)$ on an input $I$ of $Π$ is a pair $(J, Π(J))$ such that $\mathsf{dist}(I,J) \leqslant d(|I|)$ 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 practical and theoretical motivations, and compare it to the neighboring concepts of smoothed analysis, planted problems, and edition problems. Informally, we call hardness radius the ``largest'' $d$ such that Sidestep$(Π, \mathsf{dist}, d)$ is NP-hard. This framework calls for establishing the hardness radius of problems $Π$ of interest for the relevant distances $\mathsf{dist}$. We exemplify it with graph problems and two distances $\mathsf{dist}_Δ$ and $\mathsf{dist}_e$ (the edge edit distance) such that $\mathsf{dist}_Δ(G,H)$ (resp. $\mathsf{dist}_e(G,H)$) 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 $+\infty$ otherwise. We show that the decision problems Independent Set, Clique, Vertex Cover, Coloring, Clique Cover have hardness radius $n^{\frac{1}{2}-o(1)}$ for $\mathsf{dist}_Δ$, and $n^{\frac{4}{3}-o(1)}$ for $\mathsf{dist}_e$, that Hamiltonian Cycle has hardness radius 0 for $\mathsf{dist}_Δ$, and somewhere between $n^{\frac{1}{2}-o(1)}$ and $n/3$ for $\mathsf{dist}_e$, and that Dominating Set has hardness radius $n^{1-o(1)}$ for $\mathsf{dist}_e$. We leave several open questions.