Exploring the Complexity of Layout Parameters in Tournaments and Semicomplete Digraphs

Barbero, Florian; Paul, Christophe; Pilipczuk, Michał

doi:10.1145/3196276

Cited by 4 publications

(7 citation statements)

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“…due to Fradkin [18] for computing the cutwidth of a tournament. Barbero et al [4] showed that Fradkin's approach in [18] yields a polynomial time algorithm for the Optimal Linear Arrangement problem on tournaments as well. We rely on the analysis of Barbero et al in [4] to conclude that Fradkin's approach works for Minimum Directed Bisection on tournaments as well.…”

Section: :3mentioning

confidence: 99%

“…Barbero et al [4] showed that Fradkin's approach in [18] yields a polynomial time algorithm for the Optimal Linear Arrangement problem on tournaments as well. We rely on the analysis of Barbero et al in [4] to conclude that Fradkin's approach works for Minimum Directed Bisection on tournaments as well. We establish the NP-hardness of Minimum Directed Bisection on semicomplete digraphs by a reduction from the Maximum Bisection problem on directed acyclic graphs.…”

Section: :3mentioning

confidence: 99%

“…They showed that both Cutwidth and OLA admit sub-exponential FPT algorithms on semicomplete digraphs. Later, Barbero et al [4] showed that these problems are indeed NP-hard on semicomplete digraphs. We show that Minimum Directed Bisection on semicomplete digraphs admits a sub-exponential time algorithm that runs in time 2 O( √ k log k) n O(1) (Theorem 21).…”

Section: :3mentioning

confidence: 99%

“…Barbero et al [4] showed that while Cutwidth on semicomplete digraphs does not admit a polynomial kernel unless NP ⊆ coNP/poly, OLA does admit a kernel with 2k vertices. In addition, by using the submodularity property of directed cuts, Barbero et al [4] showed that Cutwidth admits a Turing kernel on semicomplete digraphs.…”

Section: :3mentioning

confidence: 99%

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A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

2021

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Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that |L| and |R| differ by at most 1 and the number of arcs from R to L is at most k. This problem, in general, is W-hard as it is known to be NP-hard even when k = 0. We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection on semicomplete digraphs is one of a handful of problems that admit sub-exponential time fixedparameter tractable algorithms. That is, we show that the problem admits a 2 O(√ k log k) n O(1) time algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use (n, k, k 2)-splitters. To the best of our knowledge, this is the first time such pseudorandom objects have been used in the design of kernels. We believe that the framework of designing kernels using splitters could be applied to more problems that admit sub-exponential time algorithms via chromatic coding. To complement the above mentioned results, we prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.

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Section: :3mentioning

confidence: 99%

Section: :3mentioning

confidence: 99%

Section: :3mentioning

confidence: 99%

Section: :3mentioning

confidence: 99%

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A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

2021

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“…Understanding the power of Turing kernelization is one of the main open research horizons in parameterized algorithmics. There is a handful of problems for which a nontrivial Turing kernelization is known [1,3,4,6,14,17,19,20,26,28]. On the other hand, there is a hierarchy of parameterized complexity classes which are conjectured not to admit polynomial Turing kernels [16].…”

Section: Introductionmentioning

confidence: 99%

Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

2019

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The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size k O(1) ?We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K 3,t -minorfree graphs. Moreover, we show that k-Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minorfree graph that does not contain a k-path, has a separation that can safely be reduced after communication with the oracle.

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Sub-exponential Time Parameterized Algorithms for Graph Layout Problems on Digraphs with Bounded Independence Number

et al. 2023

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Fradkin and Seymour (J Comb Theory Ser B 110:19–46, 2015) defined the class of digraphs of bounded independence number as a generalization of the class of tournaments. They argued that the class of digraphs of bounded independence number is structured enough to be exploited algorithmically. In this paper, we further strengthen this belief by showing that several cut problems that admit sub-exponential time parameterized algorithms (a trait uncommon to parameterized algorithms) on tournaments, including Directed Feedback Arc Set, Directed Cutwidth and Optimal Linear Arrangement, also admit such algorithms on digraphs of bounded independence number. Towards this, we rely on the generic approach of Fomin and Pilipczuk (in: Proceedings of the Algorithms—ESA 2013—21st Annual European Symposium, Sophia Antipolis, France, September 2–4, 2013, pp. 505–516, 2013), where to get the desired algorithms, it is enough to bound the number of k-cuts in digraphs of bounded independence number by a sub-exponential FPT function (Fomin and Pilipczuk bounded the number of k-cuts in transitive tournaments). Specifically, our main technical contribution is a combinatorial result that proves that the yes-instances of the problems (defined above) have a sub-exponential number of k-cuts. We prove this bound by using a combination of chromatic coding, inductive reasoning and exploiting the structural properties of these digraphs.

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Exploring the Complexity of Layout Parameters in Tournaments and Semicomplete Digraphs

Cited by 4 publications

References 29 publications

A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

Sub-exponential Time Parameterized Algorithms for Graph Layout Problems on Digraphs with Bounded Independence Number

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