Unit Interval Vertex Deletion: Fewer Vertices are Relevant

Ke, Yuping; Cao, Yixin; Ouyang, Xiating; Wang, Jianxin

doi:10.48550/arxiv.1607.01162

Cited by 5 publications

(2 citation statements)

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“…Polynomial kernels for unit interval completion [2] and unit interval vertex deletion [13] have been known for a while. Using the approximation algorithm of Theorem 1.2, we [19] recently developed an O(k 4 )-vertex kernel for unit interval vertex deletion, improving from the O(k 53 ) one of Fomin et al [13]. We conjecture that the unit interval edge deletion problem also has a small polynomial kernel.…”

Section: Discussionmentioning

confidence: 81%

Unit Interval Editing is Fixed-Parameter Tractable

Cao

2015

Automata, Languages, and Programming

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Given a graph G and integers k 1 , k 2 , and k 3 , the unit interval editing problem asks whether G can be transformed into a unit interval graph by at most k 1 vertex deletions, k 2 edge deletions, and k 3 edge additions. We give an algorithm solving this problem in time 2 O(k log k) · (n + m), where k := k 1 + k 2 + k 3 , and n, m denote respectively the numbers of vertices and edges of G. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations.Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time O(4 k · (n + m)). Another result is an O(6 k · (n + m))-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time O(6 k · n 6 ).

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Section: Discussionmentioning

confidence: 81%

Unit Interval Editing is Fixed-Parameter Tractable

Cao

2015

Automata, Languages, and Programming

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“…Coming to the question of polynomial kernels for these problems, the situation is even more grim. Until recently, the only known result was a polynomial kernel for Proper Interval Vertex Deletion: Fomin et al [19] obtained a O(k 53 ) sized polynomial kernel for Proper Interval Vertex Deletion, which has recently been improved to O(k 4 ) [27]. A dearth of further results in this area has led to the questions of kernelization complexity of Chordal Vertex Deletion and Interval Vertex Deletion becoming prominent open problems [10,13,19,24,36].…”

Section: Introductionmentioning

confidence: 99%

Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Agrawal

Lokshtanov

Misra

et al. 2018

ACM Trans. Algorithms

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Given a graph G and a parameter k, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset U ⊆ V (G) of size at most k that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size O(k 161 log 58 k), and asked whether one can design a kernel of size O(k 10 ). While we do not completely resolve this question, we design a significantly smaller kernel of size O(k 12 log 10 k), inspired by the O(k 2 )-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.Data reduction techniques are widely applied to deal with computationally hard problems in real world applications. It has been a long-standing challenge to formally express the efficiency and accuracy of these "pre-processing" procedures. The framework of parameterized complexity turns out to be particularly suitable for a mathematical analysis of pre-processing heuristics. Formally, in parameterized complexity each problem instance is accompanied by a parameter k, and we say that a parameterized problem is fixed-parameter tractable (FPT) if there is an algorithm that solves the problem in time f (k) · |I| O(1) , where |I| is the size of the input and f is a computable function of the parameter k alone. Kernelization is the subarea of parameterized complexity that deals with the mathematical analysis of pre-processing heuristics. A parameterized problem is said to admit a polynomial kernel if there is a polynomial-time algorithm (the degree of polynomial is independent of the parameter k), called a kernelization algorithm, that reduces the input instance down to an instance whose size is bounded by a polynomial p(k) in k, while preserving the answer. This reduced instance is called a p(k)-kernel for the problem. Observe that if a problem has a kernelization algorithm, then it is also has an FPT algorithm.Kernelization appears to be an interesting computational approach not only from a theoretical perspective, but also from a practical perspective. There are many real-world applications where even very simple preprocessing can be surprisingly effective, leading to significant size-reduction of the input. Kernelization is a natural tool for measuring the quality of existing preprocessing rules proposed for specific problems as well as for designing new powerful such rules. The most fundamental question in the field of kernelization is:Let Π be parameterized problem that admits an FPT algorithm. Then, does Π admit a polynomial kernel?In recent times, the study of kernelization, centred on the above question, has been one of the main areas of research in parameterized complexity, yielding many new important contributions to theory. These include general results showing that certain classes of parameterized problems have polynomial k...

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