Kernelization

Fomin, Fedor V.; Lokshtanov, Daniel; Saurabh, Saket; Zehavi, Meirav

doi:10.1017/9781107415157

Cited by 101 publications

(48 citation statements)

References 195 publications

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“…Parameterized complexity and kernelization. We refer to the books of Cygan et al [18], Downey and Fellows [22], and Fomin et al [26] for the detailed introduction to the field. Here we only briefly review the basic notions.…”

Section: Proposition 2 Every N-vertex Chordal Graph G Admits a Nice Tree Decomposition T = (T {X T } T∈v(t)mentioning

confidence: 99%

“…Every decidable parameterized problem is if and only if it admits a kernel. However, it is unlikely that every problem in has a polynomial kernel (see, e.g., [26] for the details). Still, even if a paramterized problem admits no polynomial kernel up to some complexity conjectures, sometimes we can reduce it to solving of a polynomial number of instances of the same problem such that the size of each instance is bounded by a polynomial of the parameter.…”

Section: Proposition 2 Every N-vertex Chordal Graph G Admits a Nice Tree Decomposition T = (T {X T } T∈v(t)mentioning

confidence: 99%

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Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Fomin

Golovach

2021

Algorithmica

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We study algorithmic properties of the graph class $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e . More precisely, we identify a large class of optimization problems on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e solvable in time $$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) . Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , namely, $${\textsc {Interval}}{-ke}$$ I N T E R V A L - k e and $${\textsc {Split}}{-ke}$$ S P L I T - k e graphs.

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Section: Proposition 2 Every N-vertex Chordal Graph G Admits a Nice Tree Decomposition T = (T {X T } T∈v(t)mentioning

confidence: 99%

Section: Proposition 2 Every N-vertex Chordal Graph G Admits a Nice Tree Decomposition T = (T {X T } T∈v(t)mentioning

confidence: 99%

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Fomin

Golovach

2021

Algorithmica

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“…Kernelization is the main formalization of data reduction with provable performance guarantees [30]. It has also proven to be effective in experimental studies [1,11,49].…”

Section: Kernelizationmentioning

confidence: 99%

On approximate data reduction for the Rural Postman Problem: Theory and experiments

2020

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Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and weight d of edges traversed additionally to the required ones. Thus RPP instances cannot be polynomial-time compressed to instances of size polynomial in d unless the polynomial-time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I ′ with 2b + O(c/ε) vertices in O(n 3) time so that any α-approximate solution for I ′ gives an α(1 + ε)-approximate solution for I, for any α ≥ 1 and ε > 0. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on widespread benchmark data sets as well as on two real snow plowing instances from Berlin. We also make first steps toward a PSAKS for the parameter c.

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“…We provide basic notions on parameterized complexity in the next section. For more information on parameterized complexity, see recent books [11,15,17].…”

Section: Introductionmentioning

confidence: 99%

Parameterized Pre-Coloring Extension and List Coloring Problems

Gutin¹,

Majumdar²,

Ordyniak³

et al. 2021

SIAM J. Discrete Math.

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Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V (G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λP : X → Q for X ⊆ V (G), decide whether λP can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O * (2 k )-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n − k colors for every v ∈ V (G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k 2 ) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k 2 ) colors.

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Kernelization

Cited by 101 publications

References 195 publications

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

On approximate data reduction for the Rural Postman Problem: Theory and experiments

Parameterized Pre-Coloring Extension and List Coloring Problems

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