Partition into Triangles on Bounded Degree Graphs

Rooij, Johan M. M. van; Niekerk, M.E. van Kooten; Bodlaender, Hans L.

doi:10.1007/978-3-642-18381-2_46

Cited by 15 publications

(27 citation statements)

References 7 publications

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“…twelve in bipartite graph) is N P-complete, using the Impossibility Theorem, the result follows for the first part of the corollary. In [25], the authors proved that Partition into triangles remains N P-complete for graph of maximum degree four and the desired result is obtained using Transformation 2 and Transformation 3. …”

Section: Corollarymentioning

confidence: 95%

A complexity and approximation framework for the maximization scaffolding problem

Château

Giroudeau

2015

Theoretical Computer Science

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International audienceWe explore in this paper some complexity issues inspired by the contig scaffolding problem in bioinformatics. We focus on the following problem: given an undirected graph with no loop, and a perfect matching on this graph, find a set of cycles and paths covering every vertex of the graph, with edges alternatively in the matching and outside the matching, and satisfying a given constraint on the numbers of cycles and paths. We show that this problem is NP-complete, even in planar bipartite graphs. Moreover, we show that there is no subexponential-time algorithm for several related problems unless the Exponential-Time Hypothesis fails. Lastly, we also design two polynomial-time approximation algorithms for complete graphs

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Section: Corollarymentioning

confidence: 95%

A complexity and approximation framework for the maximization scaffolding problem

Château

Giroudeau

2015

Theoretical Computer Science

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“…Note that in strong contrast to this, K 3 -Partition is linear-time solvable on graphs with maximum degree three [16]. Furthermore, P 3 -Partition is NP-complete on chordal graphs, while K 3 -Partition is polynomial-time solvable in this case [7].…”

Section: Star Partitionmentioning

confidence: 89%

“…On the downside, P k -Partition (for fixed k ≥ 3) remains NP-complete on planar bipartite graphs [9]; this hardness result generalizes to H-Partition on planar graphs for any outerplanar pattern H with at least three vertices [2]. Partitioning into triangles, that is, K 3 -Partition, is polynomial-time solvable on chordal graphs [7] and linear-time solvable on graphs of maximum degree three [16].…”

Section: Star Partitionmentioning

confidence: 90%

Star Partitions of Perfect Graphs

Bevern

Bredereck

Bulteau

et al. 2014

Automata, Languages, and Programming

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Bevern, van, R.; Bredereck, R.; Bulteau, L.; Chen, J.; Froese, V.; Niedermeier, R.; Woeginger, G.Published: 01/01/2014 Document VersionPublisher's PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract The partition of graphs into nice subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable and NP-hard cases, for example, on interval graphs, grid graphs, and bipartite permutation graphs.

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“…For instance, van Rooij et al [RKNB13] have shown that the PARTITION INTO TRIANGLES problem restricted to graphs 7. 1.…”

Section: The Easiest Np-complete Sat(·) Problemmentioning

confidence: 99%

Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

Lagerkvist¹

2015

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In this thesis we study the worst-case time complexity of the constraint satisfaction problem parameterized by a constraint language (CSP(S)), which is the problem of determining whether a conjunctive formula over S has a model. To study the complexity of CSP(S) we borrow methods from universal algebra. In particular, we consider algebras of partial functions, called strong partial clones. This algebraic approach allows us to obtain a more nuanced view of the complexity CSP(S) than possible with algebras of total functions, clones.The results of this thesis is split into two main parts. In the first part we investigate properties of strong partial clones, beginning with a classification of weak bases for all Boolean relational clones. Weak bases are constraint languages where the corresponding strong partial clones in a certain sense are extraordinarily large, and they provide a rich amount of information regarding the complexity of the corresponding CSP problems. We then proceed by classifying the Boolean relational clones according to whether it is possible to represent every relation by a conjunctive, logical formula over the weak base without needing more than a polynomial number of existentially quantified variables. A relational clone satisfying this condition is called polynomially closed and we show that this property has a close relationship with the concept of few subpowers. Using this classification we prove that a strong partial clone is of infinite order if (1) the total functions in the strong partial clone are essentially unary and (2) the corresponding constraint language is finite. Despite this, we prove that these strong partial clones can be succinctly represented with finite sets of partial functions, bounded bases, by considering stronger notions of closure than functional composition.In the second part of this thesis we apply the theory developed in the first part. We begin by studying the complexity of CSP(S) where S is a Boolean constraint language, the generalised satisfiability problem (SAT(S)). Using weak bases we prove that there exists a relation R = = = 1/3 such that SAT({R = = = 1/3 }) is the easiest NP-complete SAT(S) problem. We rule out the possibility that SAT({R = = = 1/3 }) is solvable in subexponential time unless a well-known complexity theoretical conjecture, the exponential-time hypothesis, (ETH) is false. We then proceed to study the computational complexity of two optimisation variants of the SAT(S) problem: the maximum ones problem over a Boolean constraint language S (MAX-ONES(S)) and the valued constraint satisfaction problem over a set of Boolean cost functions ∆ (VCSP(∆)). For MAX-ONES(S) we use partial clone theory and prove that MAX-ONES({R = = = 1/3 }) is the easiest NPcomplete MAX-ONES(S) problem. These algebraic techniques do not work for VCSP(∆), however, where we instead use multimorphisms to prove that MAX-CUT is the easiest NP-complete Boolean VCSP(∆) problem. Similar to the case of SAT(S) we then rule out the possibility of subexponential algorith...

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Partition into Triangles on Bounded Degree Graphs

Cited by 15 publications

References 7 publications

A complexity and approximation framework for the maximization scaffolding problem

A complexity and approximation framework for the maximization scaffolding problem

Star Partitions of Perfect Graphs

Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

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