Minimum vertex cover problems on random hypergraphs: Replica symmetric solution and a leaf removal algorithm

The minimum vertex-cover problems on random a-uniform hypergraphs are studied using two different approaches, a replica method in statistical mechanics of random systems and a leaf removal algorithm. It is found that there exists a phase transition at the critical average degree e/(alpha -1), below which a replica symmetric ansatz in the replica method holds and the algorithm estimates exactly the same solution of the problem as that by the replica method. In contrast, above the critical degree, the replica symmetric solution becomes unstable and the leaf-removal algorithm fails to estimate the optimal solution because of the emergence of a large size core. These results strongly suggest a close relation between the replica symmetry and the performance of an approximation algorithm. Critical properties of the core percolation are also examined numerically by a finite-size scaling.