| Abstract: | 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  ‐hard problems on graphs remain essentially as hard on percolated instances as they are in the worst‐case (assuming  ). We also prove hardness results for other  ‐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  and the chromatic number  are robust to percolation in the following sense. Given a graph G, let  be the graph obtained by randomly deleting edges of G with some probability  . We show that if  is small, then  remains small with probability at least 0.99. Similarly, we show that if  is large, then  remains large with probability at least 0.99. We believe these results are of independent interest. |
