11. Kernel Lower Bounds - COMP6741: Parameterized and Exact Computation

Fixed-parameter tractability Definition 2. A parameterized problem Π is fixed-parameter tractable (FPT) if there is an algorithm solving Π in time f(k) · poly(n), where n is the instance size, k is the parameter, poly is a polynomial function, and f is a computable function. Theorem 3. Let Π be a decidable parameterized problem. Π has a kernelization ⇔ Π is FPT. 2 A kernel for Hamiltonian Cycle Hamiltonian Cycle A Hamiltonian cycle of G is a subgraph of G that is a cycle on |V (G)| vertices. vc-Hamiltonian Cycle Input: A graph G = (V,E). Parameter: k = vc(G), the size of a smallest vertex cover of G. Question: Does G have a Hamiltonian cycle? Thought experiment: Imagine a very large instance where the parameter is tiny. How can you simplify such an instance? Issue: We do not actually know a vertex cover of size k.