Abstract
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 κ Q (G), plays a crucial role in quantum cryptography. In particular deciding, given a graph G and k > 0, whether κ Q (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 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.
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Cattanéo, D., Perdrix, S. (2013). Parameterized Complexity of Weak Odd Domination Problems. In: Gąsieniec, L., Wolter, F. (eds) Fundamentals of Computation Theory. FCT 2013. Lecture Notes in Computer Science, vol 8070. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40164-0_13
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DOI: https://doi.org/10.1007/978-3-642-40164-0_13
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