Computation of Hadwiger Number and Related Contraction Problems

We prove that the Hadwiger number of an n-vertex graph G (the maximum size of a clique minor in G) cannot be computed in time no(n), unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of no(n)-time algorithms (up to the ETH) for a large class of computational problems concerning edge contractions in graphs.