Abstract
In the Partial Vertex Cover (PVC) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to find a vertex subset S of size k maximizing the number of edges with at least one end-point in S. This problem is W[1]-hard on general graphs, but admits a parameterized subexponential time algorithm with running time \(2^{O(\sqrt{k})}n^{O(1)}\) on planar and apex-minor free graphs [Fomin et al. (FSTTCS 2009, IPL 2011)], and a \(k^{O(k)}n^{O(1)}\) time algorithm on bounded degeneracy graphs [Amini et al. (FSTTCS 2009, JCSS 2011)]. Graphs of bounded degeneracy contain many sparse graph classes like planar graphs, H-minor free graphs, and bounded tree-width graphs (see Fig. 1). In this work, we prove the following results:
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There are algorithms for PVC on graphs of degeneracy d with running time \(2^{O(dk)}n^{O(1)}\) and \(\left( e+ed\right) ^k 2^{o(k)} n^{O(1)}\) which are improvements on the previous \(k^{O(k)}n^{O(1)}\) time algorithm by Amini et al. [2]
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PVC admits a polynomial compression on graphs of bounded degeneracy, resolving an open problem posed by Amini et al. [2].
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Notes
- 1.
For basic definitions related to parameterized algorithms and complexity we refer to Sect. 2.1.
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Panolan, F., Yaghoubizade, H. (2022). Partial Vertex Cover on Graphs of Bounded Degeneracy. In: Kulikov, A.S., Raskhodnikova, S. (eds) Computer Science – Theory and Applications. CSR 2022. Lecture Notes in Computer Science, vol 13296. Springer, Cham. https://doi.org/10.1007/978-3-031-09574-0_18
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