Abstract
The Nemhauser and Trotter’s theorem applies to the famous Vertex Cover problem and can obtain a 2-approximation solution and a problem kernel of 2k vertices. This theorem is a famous theorem in combinatorial optimization and has been extensively studied. One way to generalize this theorem is to extend the result to the Bounded-Degree Vertex Deletion problem. For a fixed integer \(d\ge 0\), Bounded-Degree Vertex Deletion asks to delete at most k vertices of the input graph to make the maximum degree of the remaining graph at most d. Vertex Cover is a special case that \(d=0\). Fellows, Guo, Moser and Niedermeier proved a generalized theorem that implies an O(k)-vertex kernel for Bounded-Degree Vertex Deletion for \(d=0\) and 1, and for any \(\varepsilon >0\), an \(O(k^{1+\varepsilon })\)-vertex kernel for each \(d\ge 2\). In fact, it is still left as an open problem whether Bounded-Degree Vertex Deletion parameterized by k admits a linear-vertex kernel for each \(d\ge 3\). In this paper, we refine the generalized Nemhauser and Trotter’s theorem. Our result implies a linear-vertex kernel for Bounded-Degree Vertex Deletion parameterized by k for each \(d\ge 0\).
M. Xiao—Supported by NFSC of China under the Grant 61370071.
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Xiao, M. (2015). On a Generalization of Nemhauser and Trotter’s Local Optimization Theorem. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_38
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DOI: https://doi.org/10.1007/978-3-662-48971-0_38
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