A unified treatment of linked and lean tree-decompositions

Erde, Joshua

doi:10.1016/j.jctb.2017.12.001

Cited by 10 publications

(12 citation statements)

References 23 publications

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“…Seymour and Kim called directed path-decompositions satisfying (5.1), as well as two other technical conditions, 'linked'. However, when thinking about directed path-decompositions in terms of separations, perhaps a more natural concept to call 'linked' is the following (See [10]). We say a directed pathdecomposition (P, V) with P = {t 1 , .…”

Section: Linked Directed Path-decompositionsmentioning

confidence: 99%

Directed Path-Decompositions

Erde

2020

SIAM J. Discrete Math.

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Many of the tools developed for the theory of tree-decompositions of graphs do not work for directed graphs. In this paper we show that some of the most basic tools do work in the case where the model digraph is a directed path. Using these tools we define a notion of a directed blockage in a digraph and prove a min-max theorem for directed path-width analogous to the result of Bienstock, Roberston, Seymour and Thomas for blockages in graphs. Furthermore, we show that every digraph with directed path width ≥ k contains each arboresence of order ≤ k as a butterfly minor. Finally we also show that every digraph admits a linked directed path-decomposition of minimum width, extending a result of Kim and Seymour on semi-complete digraphs.• each X ∈ B is a subset of V (G) with |∂(X)| ≤ k;• if X ∈ B, Y ⊆ X and |∂(Y )| ≤ k, then Y ∈ B;

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Section: Linked Directed Path-decompositionsmentioning

confidence: 99%

Directed Path-Decompositions

Erde

2020

SIAM J. Discrete Math.

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“…GHDs of acyclic hypergraphs. ere is a related notion for Tree Decompositions called Lean Tree Decompositions (LTDs) [11,23,55].…”

Section: Widths Of Ghds E Internal Node Width (H ) Of a Ghd Focuses mentioning

confidence: 99%

“…It is broadcast to all players in K along with a counter c t ∈ [0, |K |]. 23 Initially, we set c t = 0. For any player ∈ K and j ∈ [2, |P |], de ne R be the set of tuples in R χ ( j ) mapped to by h χ ( j ) .…”

Section: G6 Hash-based Split Of Relationsmentioning

confidence: 99%

Topology Dependent Bounds For FAQs

Langberg

Shi

Jayaraman

et al. 2019

Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate eries (FAQs) studied by Abo Khamis et al. [PODS 2016] in a synchronous distributed network under the model considered by Cha opadhyay et al. [FOCS 2014, SODA 2017. Unlike the recent work on computing database queries in the Massively Parallel Computation model, in the model of Cha opadhyay et al., nodes can communicate only via private point-to-point channels and we are interested in bounds that work over an arbitrary communication topology. is model, which is closer to the well-studied CONGEST model in distributed computing and generalizes Yao's two party communication complexity model, has so far only been studied for problems that are common in the two-party communication complexity literature.is is the rst work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two special problems in this paper: Boolean Conjunctive ery (BCQ) and computing variable/factor marginals in Probabilistic Graphical Models (PGMs). We obtain tight bounds on the number of rounds needed to compute such queries as long as the underlying hypergraph of the query is O(1)-degenerate and has O(1)-arity. In particular, the O(1)-degeneracy condition covers most well-studied queries that are e ciently computable in the centralized computation model like queries with constant treewidth. ese tight bounds depend on a new notion of 'width' (namely internal-node-width) for Generalized Hypertree Decompositions (GHDs) of acyclic hypergraphs, which minimizes the number of internal nodes in a sub-class of GHDs. To the best of our knowledge, this width has not been studied explicitly in the theoretical database literature. Finally, we consider the problem of computing the product of a vector with a chain of matrices and prove tight bounds on its round complexity (over the nite eld of two elements) using a novel min-entropy based argument.

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“…The aim of Robertson and Seymour was to use Theorem 1.1 as an ingredient in their proof that graphs of bounded treewidth are well-quasi-ordered by the minor relation [RS90]. Since then, the notions of leanness and linkedness have been extensively studied and extended to several different width parameters such as θ-tree-width [CDHH14, GJ16], pathwidth [Lag98], directed path-width [KS15], DAG-width [Kin14], rank-width [Oum05], linearrankwidth [KK14], profile-and block-width [Erd18], matroid treewidth [GGW02a, Azz11,Erd18] and matroid branchwidth [GGW02a]. They have important applications, for instance in order to bound the size of obstructions for certain classes of graphs [Sey93, Lag98, KK14, GW02, GPR + 18], in well-quasi-ordering proofs [Oum08,Liu14,GGW02b], in extremal graph theory [OOT93,CRS11], and for algorithmic purpose [CKL + 18].…”

Section: Introductionmentioning

confidence: 99%

“…They have important applications, for instance in order to bound the size of obstructions for certain classes of graphs [Sey93, Lag98, KK14, GW02, GPR + 18], in well-quasi-ordering proofs [Oum08,Liu14,GGW02b], in extremal graph theory [OOT93,CRS11], and for algorithmic purpose [CKL + 18]. We refer to [Erd18] for an unified introduction to lean decompositions.…”

Section: Introductionmentioning

confidence: 99%

A Menger-like property of tree-cut width

Giannopoulou¹,

Kwon²,

Raymond³

et al. 2018

Preprint

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In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness [A Menger-like property of tree-width: The finite case. Journal of Combinatorial Theory, Series B, 48(1): 67 -76, 1990]. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan as a possible edge-version of tree decompositions [The structure of graphs not admitting a fixed immersion. Journal of Combinatorial Theory, Series B, 110:47 -66, 2015]. We show that every graph admits a tree-cut decomposition of minimum width that additionally satisfies an edge-connectivity condition analogous to Thomas' leanness. * A preliminary version of this paper appeared in the proceedings of the 36 th Symposium on Theoretical Aspects of Computer Science (STACS '19) [GKRT19].

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A unified treatment of linked and lean tree-decompositions

Cited by 10 publications

References 23 publications

Directed Path-Decompositions

Directed Path-Decompositions

Topology Dependent Bounds For FAQs

A Menger-like property of tree-cut width

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