Algorithmic Meta-theorems for Restrictions of Treewidth

Lampis, Michael

doi:10.1007/978-3-642-15775-2_47

Cited by 65 publications

(130 citation statements)

References 12 publications

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“…If C is the class of complete and edgeless graphs, then the C-cover number equals the neighborhood diversity [21], and clearly C R 1 . Figure 1 shows the relationship between the rankwidth-d cover number and some other graph invariants.…”

Section: Rank-width Coversmentioning

confidence: 99%

“…(1) The neighborhood diversity of a graph is also a rank-width-1 cover. The neighborhood diversity is known to be upper-bounded by 2 vcn(G) [21]. (2) follows immediately from the definition of rank-width-d covers.…”

Section: Rank-width Coversmentioning

confidence: 99%

“…In particular, these techniques can be applied to a wide range of graph problems parameterized by treewidth and other width parameters such as clique-width, or rank-width. Thus, in order to get polynomial kernels, structural parameters have been suggested that are somewhat weaker than treewidth, including the vertex cover number, max-leaf number, and neighborhood diversity [13,21]. The general aim is to find a parameter that admits a polynomial kernel while being as general as possible.…”

Section: Introductionmentioning

confidence: 99%

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Meta-kernelization with structural parameters

Ganian

Slivovsky

Szeider

2016

Journal of Computer and System Sciences

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Abstract. Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS'09) and of Fomin et al. (FOCS'10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the input and not the solution size. Let C be a graph class. We define the C-cover number of a graph to be a the smallest number of modules the vertex set can be partitioned into such that each module induces a subgraph that belongs to the class C. We show that each graph problem that can be expressed in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the C-cover number for any fixed class C of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean width). Many graph problems such as INDEPENDENT DOMINATING SET, c-COLORING, and c-DOMATIC NUMBER are covered by this meta-kernelization result. Our second result applies to MSO expressible optimization problems, such as MINIMUM VERTEX COVER, MINIMUM DOMINATING SET, and MAXIMUM CLIQUE. We show that these problems admit a polynomial annotated kernel with a linear number of vertices.

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Section: Rank-width Coversmentioning

confidence: 99%

Section: Rank-width Coversmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Meta-kernelization with structural parameters

Ganian

Slivovsky

Szeider

2016

Journal of Computer and System Sciences

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“…A recent paper by Lampis [123] initiated the quest for graph parameters for which the non-elementary blow-up of the constants in the running time of algorithms solving the MSO Model-Checking problem can be avoided.…”

Section: Avoiding the Non-elementary Blow-upmentioning

confidence: 99%

“…Lampis [123] considers graphs that admit a small vertex cover or a small maximum-leaf spanning tree. Both measures are typically larger than the treewidth of a graph and can be hence be understood as restrictions of treewidth, i.e., the class of bounded vertex cover number k is strictly contained in the class of graphs of bounded treewidth k. Lampis shows that for these graph classes and MSO 1 -definable properties, the non-elementary lower bounds shown by Frick and Grohe [76] can be avoided and replaced by a double-exponential dependency on k.…”

Section: Avoiding the Non-elementary Blow-upmentioning

confidence: 99%

Practical algorithms for MSO model-checking on tree-decomposable graphs

Langer

Reidl

Rossmanith

et al. 2014

Computer Science Review

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In this survey, we review practical algorithms for graph-theoretic problems that are expressible in monadic second-order logic. Monadic second-order (MSO) logic allows quantifications over unary relations (sets) and can be used to express a host of useful graph properties such as connectivity, c-colorability (for a fixed c), Hamiltonicity and minor inclusion. A celebrated theorem in this area by Courcelle states that any graph problem expressible in MSO can be solved in linear time on graphs that admit a tree-decomposition of constant width. Courcelle's Theorem has been used thus far as a theoretic tool to establish that linear-time algorithms exist for graph problems by demonstrating that the problem in question is expressible by an MSO formula. A straightforward implementation of the algorithm in the proof of Courcelle's Theorem is useless as it runs into space-explosion problems even for small values of treewidth. Of late, there have been several attempts to circumvent these problems and we review some of these in this survey. This survey also introduces the reader to the notions of tree-decompositions and the basics of monadic second order logic.

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Evangelism in social networks: Algorithms and complexity

et al. 2017

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We consider a population of interconnected individuals that, with respect to a piece of information, at each time instant can be subdivided into three (time-dependent) categories: agnostics, influenced, and evangelists. A dynamical process of information diffusion evolves among the individuals of the population according to the following rules. Initially, all individuals are agnostic. Then, a set of people is chosen from the outside and convinced to start evangelizing, i.e., to start spreading the information. When a number of evangelists, greater than a given threshold, communicate with a node v, the node v becomes influenced, whereas, as soon as the individual v is contacted by a sufficiently much larger number of evangelists, it is itself converted into an evangelist and consequently it starts spreading the information. The question is: How to choose a bounded cardinality initial set of evangelists so as to maximize the final number of influenced individuals? We prove that the problem is hard to solve, even in an approximate sense. On the positive side, we present exact polynomial time algorithms for trees and complete graphs. For general graphs, we derive exact parameterized algorithms. We also investigate the problem when the objective is to select a minimum number of evangelists capable of influencing the whole network. Our motivations to study these problems come from the areas of Viral Marketing and the analysis of quantitative models of spreading of influence in social networks.

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Algorithmic Meta-theorems for Restrictions of Treewidth

Cited by 65 publications

References 12 publications

Meta-kernelization with structural parameters

Meta-kernelization with structural parameters

Practical algorithms for MSO model-checking on tree-decomposable graphs

Evangelism in social networks: Algorithms and complexity

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