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Hypertree-depth and minors in hypergraphs

Isolde Adler, Tomáš Gavenčiak, Tereza Klimošová
2012 Theoretical Computer Science  
We introduce two new notions for hypergraphs, hypertree-depth and minors in hypergraphs.  ...  Furthermore, we show that minors in hypergraphs are 'well-behaved' with respect to hypertree-depth and other hypergraph invariants, such as generalised hypertree-depth and generalised hyperpath-width.  ...  Hypertree-depth is a hypergraph generalisation similar to generalised hypertree-width [8] and generalised hyperpath-width [10] . We propose a notion of minors in hypergraphs.  ... 
doi:10.1016/j.tcs.2012.09.007 fatcat:tc53ms3r3ng6zpe7sa75wydvcq

A Note on Directed Treewidth [article]

Sebastian Wiederrecht
2019 arXiv   pre-print
Based on this we show that a digraph has directed treewidth one if and only if its cycle hypergraph is a hypertree.  ...  cycle in D, and the directed treewidth of D.  ...  For a more in-depth introduction and more definitions we refer to [Die12] . Definition 2.1.  ... 
arXiv:1910.01826v1 fatcat:7t6hzfy3gbhmdnckzhjav5iuve

A Linear Vertex Kernel for Maximum Internal Spanning Tree [chapter]

Fedor V. Fomin, Serge Gaspers, Saket Saurabh, Stéphan Thomassé
2009 Lecture Notes in Computer Science  
Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that a hypergraph H contains a hypertree if and only if H is partition connected  ...  In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel.  ...  We stop when the resulting hypergraph H is not partitionconnected, and every contraction results in a partition-connected hypergraph.  ... 
doi:10.1007/978-3-642-10631-6_29 fatcat:ijubxsk7hfabdftqugr5orgdsy

A linear vertex kernel for maximum internal spanning tree

Fedor V. Fomin, Serge Gaspers, Saket Saurabh, Stéphan Thomassé
2013 Journal of computer and system sciences (Print)  
Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that a hypergraph H contains a hypertree if and only if H is partition connected  ...  In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel.  ...  We stop when the resulting hypergraph H is not partitionconnected, and every contraction results in a partition-connected hypergraph.  ... 
doi:10.1016/j.jcss.2012.03.004 fatcat:payufjicgrewbgoray25pw6wqi

A Linear Vertex Kernel for Maximum Internal Spanning Tree [article]

Fedor V. Fomin, Serge Gaspers, Saket Saurabh, Stéphan Thomassé
2012 arXiv   pre-print
Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that "a hypergraph H contains a hypertree if and only if H is partition connected  ...  In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel.  ...  In this case G is a 3k-vertex kernel. Otherwise proceed with Rule 2. Rule 2 Choose an arbitrary vertex v ∈ V and run a DFS (depth first search) from v.  ... 
arXiv:0907.3208v2 fatcat:23a7ddfgojha5jboseo4rtam7u

Heuristic Hypertree Decomposition

Thomas Korimort, Georg Gottlob
2003 Zenodo  
Recently, a new method called hypertree decomposition has been developed by Gottlob and others, that dominates in a precise sense all known methods from the literature.  ...  The quality of the solution is compared in the environment of a constraint solver developed by DaimlerChrylser, which can import strategies generated from hypertree decompositions and use them for the  ...  The union of hypergraph minors can be built and also the methods for computing separators are contained in the hypergraph object.  ... 
doi:10.5281/zenodo.4578376 fatcat:egyrdo3clzgnrgi5jjpbszhbhy

The monadic second-order logic of graphs XIV: uniformly sparse graphs and edge set quantifications

Bruno Courcelle
2003 Theoretical Computer Science  
This result extends to the class of uniformly k-sparse simple hypergraphs of rank at most m (for any k and m).  ...  that expresses the same property of the graphs in US k .  ...  Diestel for the proof of Proposition 1.1, and the referees for many comments resulting in simpliÿcations of proofs and improvements of the presentation.  ... 
doi:10.1016/s0304-3975(02)00578-9 fatcat:4aozazrqarbvpmald6ifup3mxm

Generalized loop-erased random walks and approximate reachability

Igor Gorodezky, Igor Pak
2012 Random structures & algorithms (Print)  
In several special cases, this algorithm perfectly samples spanning hypertrees in expected polynomial time.  ...  In this paper we extend the loop-erased random walk (LERW) to the directed hypergraph setting. We then generalize Wilson's algorithm for uniform sampling of spanning trees to directed hypergraphs.  ...  We are grateful to Carly Klivans and Jeremy Martin for helpful conversations on topological hypertrees, and to Marek Biskup, Mark Huber, Laurent Saloff-Coste, Alistair Sinclair, and Prasad Tetali for probabilistic  ... 
doi:10.1002/rsa.20460 fatcat:e4g66vwmbrdptmvo4ueuuvzziu

Fractional Covers of Hypergraphs with Bounded Multi-Intersection

Georg Gottlob, Matthias Lanzinger, Reinhard Pichler, Igor Razgon, Javier Esparza, Daniel Kráľ
2020 International Symposium on Mathematical Foundations of Computer Science  
This allows us to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is ≤ k for some constant k.  ...  Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself.  ...  We believe that an in-depth study of the connections to independent sets is merited.  ... 
doi:10.4230/lipics.mfcs.2020.41 dblp:conf/mfcs/GottlobLPR20 fatcat:c6gk4miioveujj3v57cgzehzca

Fractional Covers of Hypergraphs with Bounded Multi-Intersection [article]

Georg Gottlob, Matthias Lanzinger, Reinhard Pichler, Igor Razgon
2020 arXiv   pre-print
This allows us to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is $\leq k$ for some constant $k$.  ...  Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself.  ...  We believe that an in-depth study of the connections to independent sets is merited.  ... 
arXiv:2007.01830v2 fatcat:jyswprlf25fzxo2xsj5dj7o5lu

A New Class of Upper Bounds on the Log Partition Function

M.J. Wainwright, T.S. Jaakkola, A.S. Willsky
2005 IEEE Transactions on Information Theory  
Our derivation is based on concepts from convex duality and information geometry: in particular, it exploits mixtures of distributions in the exponential domain, and the Legendre mapping between exponential  ...  The analysis extends naturally to convex combinations of hypertree-structured distributions, thereby establishing links to Kikuchi approximations and variants.  ...  Nati Srebro for helpful discussions on hypertrees; and to the anonymous referees for their comments and constructive criticism that helped to improve the presentation of the work.  ... 
doi:10.1109/tit.2005.850091 fatcat:gj3suuqbcnawdlaipub25a7jt4

A Join-Based Hybrid Parameter for Constraint Satisfaction [chapter]

Robert Ganian, Sebastian Ordyniak, Stefan Szeider
2019 Lecture Notes in Computer Science  
It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw).  ...  Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account.  ...  We denote by H[V ′ ] the hypergraph induced on the vertices in V ′ ⊆ V , i.e., the hypergraph with vertex set V ′ and edge set { e ∩ V ′ : e ∈ E }.  ... 
doi:10.1007/978-3-030-30048-7_12 fatcat:dydl7jwmxfcjvn5mp5wrsbyi3a

Tractable hypergraph properties for constraint satisfaction and conjunctive queries

Dániel Marx
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H (which is a strictly more general property than bounded fractional hypertree width) implies  ...  Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H.  ...  Observe that ψ is a minor mapping if and only if it has vertex depth 1.  ... 
doi:10.1145/1806689.1806790 dblp:conf/stoc/Marx10 fatcat:rue3f6bjjjgszgs37fq5bzitte

Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries

Dániel Marx
2013 Journal of the ACM  
Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H (which is a strictly more general property than bounded fractional hypertree width) implies  ...  Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H.  ...  Observe that ψ is a minor mapping if and only if it has vertex depth 1.  ... 
doi:10.1145/2535926 fatcat:z663q2gyc5a3fp4r26fnp4ivgi

Hypertree Decompositions and Tractable Queries

Georg Gottlob, Nicola Leone, Francesco Scarcello
2002 Journal of computer and system sciences (Print)  
In order to circumvent this difficulty, we introduce the new concept of hypertree decomposition of a query and the corresponding notion of hypertree width.  ...  We prove: (a) for each k, the class of queries with query width bounded by k is properly contained in the class of queries whose hypertree width is bounded by k; (b) unlike query width, constant hypertree-width  ...  The treewidth of a graph is a well-known measure of its tree-likeness introduced by Robertson and Seymour in their work on graph minors [34] .  ... 
doi:10.1006/jcss.2001.1809 fatcat:eixkkrj5zzbp7gblcvergbzzbu
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