Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity

Knop, Dušan; Koutecký, Martin; Masařík, Tomáš; Toufar, Tomáš

doi:10.1007/978-3-319-68705-6_26

Cited by 14 publications

(15 citation statements)

References 38 publications

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“…Similarly, Fiala et al [FGK + 17] shows that the Uniform Channel Assignment problem is (triple-exponential) fixed-parameter tractable parameterized by neighborhood diversity and the largest edge weight. Ganian and Obdržálek [GO16] and later Knop et al [KKMT17] study extensions of the MSO logic, and provide fixed-parameter algorithms for their model checking on graphs of bounded neighborhood diversity. These algorithms again use combinatorial pre-n-folds under the hood.…”

Section: Miscellaneousmentioning

confidence: 99%

Combinatorial n-fold integer programming and applications

2019

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Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra solves ILPs in time that is exponential only in the dimension of the program, and polynomial in the size of the ILP. That algorithm became a ubiquitous tool in the design of fixedparameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problem by some parameter. However, in many cases using Lenstra's algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension cannot easily express problems involving many different costs.Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a singleexponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both.Unlike Lenstra's algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs, existence of an augmenting step implies existence of a "local" augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.1 Kannan's algorithm in parameterized complexity: they modeled Closest String with k input strings as an ILP of dimension k O(k) , and thereby concluded with the first fixed-parameter algorithm for Closest String. This success led Niedermeier [Nie04] to propose in his book:[...] It remains to investigate further examples besides Closest String where the described ILP approach turns out to be applicable. More generally, it would be interesting to discover more connections between fixed-parameter algorithms and (integer) linear programming.Since then, many more applications of Lenstra's and Kannan's algorithm for parameterized problems have been proposed. However, essentially all of them [BFN + 15, DS12, FLM + 08, HR15, Lam12, MW15] share a common trait with the algorithm for Closest String: they have a doubly-exponential run time dependence on the parameter. Moreover, it is difficult to find ILP formulations with small dimension for problems whose input contains many objects with varying cost functions, such as in Swap Bribery [BCF + 14, Challenge #2].

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Section: Miscellaneousmentioning

confidence: 99%

Combinatorial n-fold integer programming and applications

2019

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“…We start from two recently introduced parameters: modular-width [22] and neighborhood diversity [31]. Both parameters received much attention [1,2,5,7,12,17,18,21,24,25,29] also due to their property of being computable in polynomial time [22,31].…”

Section: Introductionmentioning

confidence: 99%

Iterated Type Partitions

Cordasco

Gargano

Rescigno

2020

Lecture Notes in Computer Science

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This paper introduces a novel parameter, called iterated type partition, that can be computed in polynomial time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parametrized by the iterated type partition. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parametrized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity. Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition, which enables us to find optimal solutions for several graph problems. While the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. As an example, in this paper, we give an algorithm for the Dominating Set problem that outputs an optimal set in time O(2 t + poly(n)), where n and t are the size and the iterated type partition of the input graph, respectively.

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“…MSO has been extended in several ways to increase its expressive power while retaining FPT (or at least XP) membership of model checking when parameterized by treewidth. For overviews, we refer the reader to Knop, Koutecký, Masarík, and Toufar (2017) and Langer, Reidl, Rossmanith, and Sikdar (2014). On the other hand, also our results on (connection-) guarded ASP can be extended to more powerful languages.…”

Section: Discussionmentioning

confidence: 85%

The Impact of Treewidth on Grounding and Solving of Answer Set Programs

Bliem

Morak

Moldovan

et al. 2020

jair

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In this paper, we aim to study how the performance of modern answer set programming (ASP) solvers is influenced by the treewidth of the input program and to investigate the consequences of this relationship. We first perform an experimental evaluation that shows that the solving performance is heavily influenced by treewidth, given ground input programs that are otherwise uniform, both in size and construction. This observation leads to an important question for ASP, namely, how to design encodings such that the treewidth of the resulting ground program remains small. To this end, we study two classes of disjunctive programs, namely guarded and connection-guarded programs. In order to investigate these classes, we formalize the grounding process using MSO transductions. Our main results show that both classes guarantee that the treewidth of the program after grounding only depends on the treewidth (and the maximum degree, in case of connection-guarded programs) of the input instance. In terms of parameterized complexity, our findings yield corresponding FPT results for answer-set existence for bounded treewidth (and also degree, for connection-guarded programs) of the input instance. We further show that bounding treewidth alone leads to NP-hardness in the data complexity for connection-guarded programs, which indicates that the two classes are fundamentally different. Finally, we show that for both classes, the data complexity remains as hard as in the general case of ASP.

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Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity

Cited by 14 publications

References 38 publications

Combinatorial n-fold integer programming and applications

Combinatorial n-fold integer programming and applications

Iterated Type Partitions

The Impact of Treewidth on Grounding and Solving of Answer Set Programs

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