Lecture 8 : Ladner ’ s theorem , self - reducibility

1 Hamiltonian Cycle “Non-local” problems can also be NP-hard. Recall that a Hamilton cycle is one where all vertices is visited exactly once. Name: HC Input: A graph G. Output: Does G contain a Hamiltonian cycle? In other words, HC is asking whether the longest cycle in G has length n. This problem is “non-local” in the sense that the constraint involves all vertices (a.k.a. global). Theorem 1. HC is NP-hard. Proof. We will reduce from VC. Given G = (V,E), an instance of VC, we construct a graph G′. For each e ∈ E, we replace it by a “gadget” shown in Figure 1. The gadget in Figure 1 contains three possible