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Parametrized Complexity of Weak Odd Domination Problems

Given a graph $G=(V,E)$, a subset $B\subseteq V$ of vertices is a weak odd dominated (WOD) set if there exists $D \subseteq V {\setminus} B$ 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 $κ_Q(G)$, plays a crucial role in quantum cryptography. In particular deciding, given a graph $G$ and $k>0$, whether $κ_Q(G)\le k$ is of practical interest in the design of graph-based quantum secret sharing schemes. The decision problems associated with the quantities $κ$, $κ'$ and $κ_Q$ 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 $κ_Q$, $κ$ and $κ'$ admit a constant factor approximation algorithm, and that $κ$ and $κ'$ have no polynomial approximation scheme unless P=NP.