On percolation and -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 complexity of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical NP-hard problems on graphs remain essentially as hard on percolated instances as they are in the worst-case (assuming NP not subset of BPP). We also prove hardness results for other NP-hard problems such as Constraint Satisfaction Problems and Subset-Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number alpha(G) and the chromatic number chi(G) are robust to percolation in the following sense. Given a graph G, let G' be the graph obtained by randomly deleting edges of G with some probability p is an element of(0, 1). 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. We believe these results are of independent interest.