Answering Related Questions

TL;DR

The paper introduces the Sidestep problem framework to analyze the complexity of solving related questions within a certain distance, exemplified through graph problems and distances like maximum degree and edge edit distance.

Contribution

It formalizes the Sidestep problem, defines the hardness radius concept, and computes this radius for various classical graph problems under specific distance metrics.

Findings

Independent Set, Clique, Vertex Cover, Coloring, Clique Cover have hardness radius ~n^{1/2} for maximum degree distance.

These problems have a hardness radius of ~n^{4/3} for edge edit distance.

Hamiltonian Cycle has zero hardness radius for maximum degree distance.

Abstract

We introduce the meta-problem Sidestep$(\Pi, \mathsf{dist}, d)$ for a problem $\Pi$, a metric $\mathsf{dist}$ over its inputs, and a map $d: \mathbb N \to \mathbb R_+ \cup \{\infty\}$. A solution to Sidestep$(\Pi, \mathsf{dist}, d)$ on an input $I$ of $\Pi$ is a pair $(J, \Pi(J))$ such that $\mathsf{dist}(I,J) \leqslant d(|I|)$ and $\Pi(J)$ is a correct answer to $\Pi$ 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 to the neighboring concepts of smoothed analysis, certified algorithms, planted problems, edition problems, and approximation algorithms. Informally, we call hardness radius the ``largest'' $d$ such that Sidestep$(\Pi, \mathsf{dist}, d)$ is NP-hard. This framework calls for establishing the hardness radius of problems $\Pi$ of interest for the relevant distances $\mathsf{dist}$. We exemplify it with graph problems and two distances $\mathsf{dist}_\Delta$ and $\mathsf{dist}_e$ (the edge edit distance) such that $\mathsf{dist}_\Delta(G,H)$ (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 $\mathsf{dist}_\Delta$, and $n^{\frac{4}{3}-o(1)}$ for $\mathsf{dist}_e$, that Hamiltonian Cycle has hardness radius 0 for $\mathsf{dist}_\Delta$, 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.