Abstract | ||
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We obtain new polynomial kernels and compression algorithms for Path Cover and Cycle Cover, the well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Our choice of parameterization is strongly influenced by the work of Bir\'o, Hujter, and Tuza, who in 1992 introduced H-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi-)graph H. In this work, we turn to proper H-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph H is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results. Path Cover admits a kernel of size O (parallel to H parallel to(8)), where parallel to H parallel to is the size of graph H. In other words, we design an algorithm that for an n-vertex graph G and integer k \geq 1, in time polynomial in n and parallel to H parallel to, outputs a graph G\prime of size \scrO (parallel to H parallel to(8)) and k\prime \leq | V (G' such that the vertex set of G is coverable by k vertex-disjoint paths if and only if the vertex set of G' is coverable by k' vertex-disjoint paths. Hamiltonian Cycle admits a kernel of size O (parallel to H parallel to(8)). Cycle Cover admits a polynomial kernel. We prove it by providing a compression of size O (parallel to H parallel to(10)) into another NP-complete problem, namely, Prize Collecting Cycle Cover, that is, we design an algorithm that, in time polynomial in n and parallel to H parallel to, outputs an equivalent instance of Prize Collecting Cycle Cover of sizeO (parallel to H parallel to(10)). In all our algorithms we assume that a proper H-decomposition is given as a part of the input. |
Year | DOI | Venue |
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2021 | 10.1137/19M1299001 | SIAM JOURNAL ON DISCRETE MATHEMATICS |
Keywords | DocType | Volume |
cycle cover, path cover, proper H-graphs, kernelization | Journal | 35 |
Issue | ISSN | Citations |
2 | 0895-4801 | 0 |
PageRank | References | Authors |
0.34 | 0 | 5 |
Name | Order | Citations | PageRank |
---|---|---|---|
steven chaplick | 1 | 76 | 16.91 |
Fedor V. Fomin | 2 | 3139 | 192.21 |
Petr A. Golovach | 3 | 0 | 0.68 |
Dusan Knop | 4 | 47 | 17.31 |
Peter Zeman | 5 | 6 | 3.14 |