scholarly journals Two Short Proofs Concerning Tree-Decompositions

2002 ◽  
Vol 11 (6) ◽  
pp. 541-547 ◽  
Author(s):  
PATRICK BELLENBAUM ◽  
REINHARD DIESTEL
Keyword(s):  
Duality Theorem ◽  
Finite Graph ◽  
Tree Decomposition ◽  
Tree Width ◽  
Tree Decompositions

We give short proofs of the following two results: Thomas's theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width; and the ‘tree-width duality theorem’ of Seymour and Thomas, that the tree-width of a finite graph is exactly one less than the largest order of its brambles.

AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

2000 ◽  
Vol 11 (03) ◽  
pp. 365-371 ◽  
Author(s):  
LJUBOMIR PERKOVIĆ ◽  
BRUCE REED
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Tree Decomposition ◽  
Linear Time Algorithm ◽  
Input Graph ◽  
Primary Motivation ◽  
Small Width ◽  
Tree Decompositions ◽  
Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

scholarly journals Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning

2017 ◽  
Vol 58 ◽  
pp. 829-858 ◽  
Author(s):  
Michael Abseher ◽  
Nysret Musliu ◽  
Stefan Woltran
Keyword(s):  
Machine Learning ◽  
Dynamic Programming ◽  
Tree Decomposition ◽  
Machine Learning Techniques ◽  
Problem Instance ◽  
Actual Problem ◽  
Learning Techniques ◽  
Tree Decompositions ◽  
Selection Of ◽  
General Method

Dynamic Programming (DP) over tree decompositions is a well-established method to solve problems - that are in general NP-hard - efficiently for instances of small treewidth. Experience shows that (i) heuristically computing a tree decomposition has negligible runtime compared to the DP step; and (ii) DP algorithms exhibit a high variance in runtime when using different tree decompositions; in fact, given an instance of the problem at hand, even decompositions of the same width might yield extremely diverging runtimes. We thus propose here a novel and general method that is based on selection of the best decomposition from an available pool of heuristically generated ones. For this purpose, we require machine learning techniques that provide automated selection based on features of the decomposition rather than on the actual problem instance. Thus, one main contribution of this work is to propose novel features for tree decompositions. Moreover, we report on extensive experiments in different problem domains which show a significant speedup when choosing the tree decomposition according to this concept over simply using an arbitrary one of the same width.

scholarly journals Linked Tree-Decompositions of Infinite Represented Matroids

2021 ◽  
Author(s):  
◽  
Jeffrey Donald Azzato
Keyword(s):  
Local Structure ◽  
Infinite Graphs ◽  
Connected Components ◽  
Finite Width ◽  
Graph Minors ◽  
Finite Tree ◽  
Tree Width ◽  
Tree Decompositions ◽  
Branch Decompositions ◽  
Quasi Ordering

<p>It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids.  Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width.</p>

scholarly journals Linked Tree-Decompositions of Infinite Represented Matroids

2021 ◽  
Author(s):  
◽  
Jeffrey Donald Azzato
Keyword(s):  
Local Structure ◽  
Infinite Graphs ◽  
Connected Components ◽  
Finite Width ◽  
Graph Minors ◽  
Finite Tree ◽  
Tree Width ◽  
Tree Decompositions ◽  
Branch Decompositions ◽  
Quasi Ordering

<p>It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids.  Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width.</p>

scholarly journals CONTEXT-FREE GROUPS AND THEIR STRUCTURE TREES

2013 ◽  
Vol 23 (03) ◽  
pp. 611-642 ◽  
Author(s):  
VOLKER DIEKERT ◽  
ARMIN WEIß
Keyword(s):  
Free Groups ◽  
Finite Graph ◽  
Fundamental Result ◽  
Direct Construction ◽  
Finite Tree ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Structure Tree ◽  
Tree Width ◽  
Context Free

Let Γ be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with finitely many orbits and finite vertex stabilizers. Moreover, the tree is defined directly in terms of the structure tree of optimally nested cuts of Γ. Once the tree is constructed, Bass–Serre theory yields that G is virtually free. This approach simplifies the existing proofs for the fundamental result of Muller and Schupp that characterizes context-free groups as finitely generated (f.g.) virtually free groups. Our construction avoids the explicit use of Stallings' structure theorem and it is self-contained. We also give a simplified proof for an important consequence of the structure tree theory by Dicks and Dunwoody which has been stated by Thomassen and Woess. It says that a f.g. group is accessible if and only if its Cayley graph is accessible.

scholarly journals D-FLAT: Declarative problem solving using tree decompositions and answer-set programming

2012 ◽  
Vol 12 (4-5) ◽  
pp. 445-464 ◽  
Author(s):  
BERNHARD BLIEM ◽  
MICHAEL MORAK ◽  
STEFAN WOLTRAN
Keyword(s):  
Dynamic Programming ◽  
Dynamic Programming Algorithm ◽  
Answer Set Programming ◽  
Tree Decomposition ◽  
Programming Algorithm ◽  
Technical Details ◽  
Local Solutions ◽  
Tree Decompositions ◽  
Programming Algorithms ◽  
Answer Set

AbstractIn this work, we propose Answer-Set Programming (ASP) as a tool for rapid prototyping of dynamic programming algorithms based on tree decompositions. In fact, many such algorithms have been designed, but only a few of them found their way into implementation. The main obstacle is the lack of easy-to-use systems which (i) take care of building a tree decomposition and (ii) provide an interface for declarative specifications of dynamic programming algorithms. In this paper, we present D-FLAT, a novel tool that relieves the user of having to handle all the technical details concerned with parsing, tree decomposition, the handling of data structures, etc. Instead, it is only the dynamic programming algorithm itself which has to be specified in the ASP language. D-FLAT employs an ASP solver in order to compute the local solutions in the dynamic programming algorithm. In the paper, we give a few examples illustrating the use of D-FLAT and describe the main features of the system. Moreover, we report experiments which show that ASP-based D-FLAT encodings for some problems outperform monolithic ASP encodings on instances of small treewidth.

scholarly journals Tree-width and large grid minors in planar graphs

2011 ◽  
Vol Vol. 13 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Alexander Grigoriev
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Tree Decomposition ◽  
International Audience ◽  
Tree Width ◽  
Grid Minor ◽  
Large Grid

Graphs and Algorithms International audience We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g - 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n(2) log n) time.

scholarly journals Constructing tree decompositions of graphs with bounded gonality

2021 ◽  
Author(s):  
Hans L. Bodlaender ◽  
Josse van Dobben de Bruyn ◽  
Dion Gijswijt ◽  
Harry Smit
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Effective Divisor ◽  
Tree Decomposition ◽  
Constructive Proof ◽  
Tree Decompositions

AbstractIn this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality.

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