On Percolation and $NP$-Hardness

The edge-percolation and vertex-percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational hardness of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical N P-hard graph problems remain essentially as hard on percolated instances as they are in the worst-case (assuming NP !subseteq BPP). We also prove hardness results for other NP-hard problems such as Constraint Satisfaction Problems, where random deletions are applied to clauses or variables. We focus on proving the hardness of the Maximum Independent Set problem and the Graph Coloring problem on percolated instances. To show this we establish the robustness of the corresponding parameters alpha(.) and Chi(.) to percolation, which may be of independent interest. Given a graph G, let G' be the graph obtained by randomly deleting edges of G. We show that if alpha(G) is small, then alpha(G') remains small with probability at least 0.99. Similarly, we show that if Chi(G) is large, then Chi(G') remains large with probability at least 0.99.