Parameterized Complexity of Weak Odd Domination Problems

Given a graph G=(V,E), a subset B⊆V of vertices is a weak odd dominated (WOD) set if there exists D⊆V∖B such that every vertex in B has an odd number of neighbours in D. κ(G) denotes the size of the largest WOD set, and κ′(G) the size of the smallest non-WOD set. The maximum of κ(G) and |V|−κ′(G), denoted κQ(G), plays a crucial role in quantum cryptography. In particular deciding, given a graph G and k>0, whether κQ(G)≤k is of practical interest in the design of graph-based quantum secret sharing schemes. The decision problems associated with the quantities κ, κ′ and κQ are known to be NP-Complete. In this paper, we consider the approximation of these quantities and the parameterized complexity of the corresponding problems. We mainly prove the fixed-parameter intractability (W[1]-hardness) of these problems. Regarding the approximation, we show that κQ, κ and κ′ admit a constant factor approximation algorithm, and that κ and κ′ have no polynomial approximation scheme unless P=NP.