Search-Space Reduction Via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909170367725568 |
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| author | Jansen, Bart M. P. Verhaegh, Ruben F. A. |
| author_facet | Jansen, Bart M. P. Verhaegh, Ruben F. A. |
| contents | For an optimization problem $Π$ on graphs whose solutions are vertex sets, a vertex $v$ is called $c$-essential for $Π$ if all solutions of size at most $c \cdot OPT$ contain $v$. Recent work showed that polynomial-time algorithms to detect $c$-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size $k$ of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for $3$-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected $n$-vertex graph $G$ in time $2^{O(\ell^3)} \cdot n^{O(1)}$, where $\ell$ is the number of vertices in an optimal solution that are not $3$-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of $c$, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that ($2-\varepsilon$)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects $2$-essential vertices is best-possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09769 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Search-Space Reduction Via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion Jansen, Bart M. P. Verhaegh, Ruben F. A. Data Structures and Algorithms For an optimization problem $Π$ on graphs whose solutions are vertex sets, a vertex $v$ is called $c$-essential for $Π$ if all solutions of size at most $c \cdot OPT$ contain $v$. Recent work showed that polynomial-time algorithms to detect $c$-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size $k$ of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for $3$-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected $n$-vertex graph $G$ in time $2^{O(\ell^3)} \cdot n^{O(1)}$, where $\ell$ is the number of vertices in an optimal solution that are not $3$-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of $c$, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that ($2-\varepsilon$)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects $2$-essential vertices is best-possible. |
| title | Search-Space Reduction Via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2404.09769 |