On Percolation and $NP$-Hardness

TL;DR

This paper investigates how the computational hardness of classical NP-hard problems persists under random deletions of edges, vertices, or clauses, showing that hardness remains unless NP is contained in BPP for certain deletion probabilities.

Contribution

It establishes that the NP-hardness of problems like Coloring, Vertex-Cover, and Hamiltonicity is robust under random deletions, extending worst-case hardness results to probabilistic settings.

Findings

Hardness persists for deletion probability p > 1/n^{1-ε}

Results apply to graph problems, CSPs, and Subset-Sum

Hardness remains unless NP ⊆ BPP

Abstract

We consider the robustness of computational hardness of problems whose input is obtained by applying independent random deletions to worst-case instances. For some classical $NP$-hard problems on graphs, such as Coloring, Vertex-Cover, and Hamiltonicity, we examine the complexity of these problems when edges (or vertices) of an arbitrary graph are deleted independently with probability $1-p > 0$. We prove that for $n$-vertex graphs, these problems remain as hard as in the worst-case, as long as $p > \frac{1}{n^{1-\epsilon}}$ for arbitrary $\epsilon \in (0,1)$, unless $NP \subseteq BPP$. We also prove hardness results for Constraint Satisfaction Problems, where random deletions are applied to clauses or variables, as well as the Subset-Sum problem, where items of a given instance are deleted at random.