Review of Vertex Cover

for a constant c ∈ (1, 2). Using more clever backtracking, one can develop even more complex recurrences and a running time of O(1.27 · poly(n)). For the minimum vertex cover problem, we can therefore solve it on arbitrary graphs in time O∗(1.27n).1 In fact, one can get a slightly better running time for arbitrary graphs, via the following trick. Notice that if the vertex cover is of size at least n − δn for some δ < 1/2, we can solve the problem in O∗( ( n δn ) ) time, by trying all vertex subsets of size at least n− δn. This bound is at most O∗(2H(δ)n), where H is the binary entropy function, i.e.,