Unit interval editing is fixed-parameter tractable

Cao, Yixin

doi:10.1016/j.ic.2017.01.008

Cited by 15 publications

(17 citation statements)

References 24 publications

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“…This way, we construct X of size O(k 3 ) . Then we construct F of size O(k 14 ) and define G . Here again, we use the technique of Frank and Tardos [30] to compress the weights.…”

Section: Independent Set On Interval−kementioning

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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“…This way, we construct X of size O(k 3 ) . Then we construct F of size O(k 14 ) and define G . Here again, we use the technique of Frank and Tardos [30] to compress the weights.…”

Section: Independent Set On Interval−kementioning

confidence: 99%

“…By Lemma 12, G * has O(k 14 ) vertices, that is, the size of G * is bounded by a polynomial of the parameter. To complete the construction of the compressed instance, it remains to reduce the weights of vertices.…”

Section: Lemma 9 Reduction Rule 3 Is Safementioning

confidence: 99%

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Fomin

Golovach

2021

Algorithmica

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“…For example, the statement "threshold graphs are self-complementary" is incorrect, because most threshold graphs are not isomorphic to their complements, though the later are necessarily threshold graphs. Our final remarks are on the approximation algorithms, for which we are concerned with those not shown to be in P. All of them have constant-ratio approximations, which follow from either [7,8] or the general observation of Lund and Yannakakis [28]. On the other hand, none of the NP-complete problems admits a polynomial-time approximation scheme.…”

Section: Introductionmentioning

confidence: 99%

“…(We defer their definitions to the next section.) Many algorithms have been developed for vertex deletion problems to chordal graphs and its subclasses,-most notably (unit) interval graphs, cluster graphs, and split graphs; see, e.g., [17,4,10,9,8,34,12,25, 1] for a partial list. After the long progress of algorithmic achievements, some natural questions arise: What is the complexity of transforming a chordal graph to a (unit) interval graph, a cluster graph, a split graph, or a member of some other subclass of chordal graphs?…”

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confidence: 99%

“…Unfortunately, we have to leave the complexity of some problems open, particularly CHORDAL → CLUSTER, CHORDAL → UNIT INTERVAL, and INTERVAL → UNIT INTERVAL. Our final remarks are on the approximation algorithms, for which we are concerned with those not shown to be in P. All of them have constant-ratio approximations, which follow from either [7,8] or the general observation of Lund and Yannakakis [28]. On the other hand, none of the NP-complete problems admits a polynomial-time approximation scheme.…”

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Vertex deletion problems on chordal graphs

Cao

Otachi

et al. 2018

Theoretical Computer Science

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Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yannakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. IntroductionGenerally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices. Many classic optimization problems belong to the family of vertex deletion problems, and their algorithms and complexity have been intensively studied. For example, the clique problem and the independent set problem are nothing but the vertex deletion problems to complete graphs and to edgeless graphs respectively. Most interesting graph properties are hereditary: If a graph satisfies this property, then so does every induced subgraph of it. For all the vertex deletion problems to hereditary graph classes, Lewis and Yannakakis [27] have settled their complexity once and for all with a dichotomy result: They are either NP-hard or trivial. Thereafter algorithmic efforts were mostly focused on the nontrivial ones, and the major approaches include approximation algorithms [28], parameterized algorithms [6], and exact algorithms [15].Chordal graphs make one of the most important graph classes. Together with many of its subclasses, it has played important roles in the development of structural graph theory. (We defer their definitions to the next section.) Many algorithms have been developed for vertex deletion problems to chordal graphs and its subclasses,-most notably (unit) interval graphs, cluster graphs, and split graphs; see, e.g., [17,4,10,9,8,34,12,25, 1] for a partial list. After the long progress of algorithmic achievements, some natural questions arise: What is the complexity of transforming a chordal graph to a (unit) interval graph, a cluster graph, a split graph, or a member of some other subclass of chordal graphs? It is quite surprising that this type of problems has not been systematically studied, save few concrete results, e.g., the polynomial-time algorithms for the clique problem, the independent set problem, and the feedback vertex set problem (the object class being forests) [21,33].The same question can be asked for other pair of source and object graph classes. The most important source classes include planar graphs [20,18,16], bipartite graphs [32], and degree-bounded graphs [19]. As one may expect, with special properties imposed on input graphs, the problems become easier, and some of them may not remain NP-hard. Unfortunately, a clear-cut answer to them seems very unlikely, since their complexity would...

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