Number of Guards Needed by a Museum: A Phase Transition in Vertex Covering of Random Graphs

In this Letter we study the $\mathrm{NP}$-complete vertex cover problem on finite connectivity random graphs. When the allowed size of the cover set is decreased, a discontinuous transition in solvability and typical-case complexity occurs. This transition is characterized by means of exact numerical simulations as well as by analytical replica calculations. The replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity $e$, where replica symmetry becomes locally unstable.