Efficient parallel algorithms for tree-decomposition and related problems
22 Oct 1990-pp 173-182
TL;DR: The sequential time complexity of the tree-composition problem for fixed w is improved, and some implications for this improvement are stated.
Abstract: An efficient parallel algorithm for the tree-decomposition problem for fixed width w is presented. The algorithm runs in time O(log/sup 3/ n) and uses O(n) processors on a concurrent-read, concurrent-write parallel random access machine (CRCW PRAM). This result can be used to construct efficient parallel algorithms for three important classes of problems: MS (monadic second-order) properties, linear EMS (extended monadic second-order) extremum problems, and enumeration problems for MS properties, for graphs of tree width at most w. The sequential time complexity of the tree-composition problem for fixed w is improved, and some implications for this improvement are stated. >
Citations
More filters
01 Jun 1993
TL;DR: Every minor-closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm that determines whether the treewidth of G is at most k, and if so, finds a treedecomposition of G withtreewidth at mostK.
Abstract: In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G=(V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. Another consequence is that a similar result holds when we look instead for path-decompositions with pathwidth at most some constant $k$.
727 citations
TL;DR: For all constantsk,l, explicit algorithms that, given a graphG=(V,E) with a tree-decomposition ofG with treewidth with useO(|V|) time are given, which do not rely on non-constructive reasoning and are single exponential inkandl.
306 citations
Cites background from "Efficient parallel algorithms for t..."
...[31] J. Lagergren and S. Arnborg....
[...]
...Results of a similar nature as ours were independently obtained by Lagergren and Arnborg [31] and by Abrahamson and Fellows [IJ....
[...]
...Lagergren [30J gives a parallel algorithm that uses O(10g3 n) time and O( n) processors on a CRCW PRAM....
[...]
...[30] J. Lagergren....
[...]
...[5] S. Arnborg, J. Lagergren, and D. Seese....
[...]
TL;DR: By using monadic second-order logic and semiring homomorphisms, this work describes in a single formalism a large class of functions on graphs that can be computed recursively on the derivation trees of these graphs.
267 citations
TL;DR: A heuristic method for finding branch decompositions based on the eigenvector technique for finding graph separators is described; this is used as a tool to obtain high-quality tours for the traveling salesman problem by merging collections of tours produced by standard traveling salesman heuristics.
Abstract: Robertson and Seymour introduced branch-width as a new connectivity invariant of graphs in their proof of the Wagner conjecture. Decompositions based on this invariant provide a natural framework for implementing dynamic-programming algorithms to solve graph optimization problems. We describe a heuristic method for finding branch decompositions; the method is based on the eigenvector technique for finding graph separators. We use this as a tool to obtain high-quality tours for the traveling salesman problem by merging collections of tours produced by standard traveling salesman heuristics.
189 citations
Cites background or methods from "Efficient parallel algorithms for t..."
...This approximation algorithm was improved in Matous̆ek and Thomas (1991), Lagergren (1990), Reed (1992), Bodlaender (1996), and Bodlaender and Kloks (1996), but all of these methods are either 236 INFORMS Journal on Computing/Vol. 15, No. 3, Summer 2003 D ow nl oa de d fr om in fo rm s. or g by…...
[...]
...For instance, in Robertson and Seymour (1991) there is a fast algorithm to estimate branch-width, within an error factor of three (that is, it would decide either that a graph has branch-width at least 10 or it would find a branch-decomposition of width at most 30), but this method is of little value in a practical implementation....
[...]
...In the classic work of Lin and Kernighan (1973), pseudo-random starting tours are used to permit repeated application of their localsearch procedure....
[...]
TL;DR: This work describes an algorithm which will produce, from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth ≤ k, a set of rewrite rules that reduces any member of the class to one of finitely many graphs, in a number of steps bounded by the size of the graph.
Abstract: We show how membership in classes of graphs definable in monadic second order logic and of bounded treewidth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that we describe an algorithm which will produce, from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth ≤ k, a set of rewrite rules that reduces any member of the class to one of finitely many graphs, in a number of steps bounded by the size of the graph. This reduction system corresponds to an algorithm that runs in time linear in the size of the graph.
187 citations
References
More filters
Book•
01 Jan 1976
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
7,497 citations
5,837 citations
TL;DR: An invariant of graphs called the tree-width is introduced, and used to obtain a polynomially bounded algorithm to test if a graph has a subgraph contractible to H, where H is any fixed planar graph.
1,726 citations
TL;DR: This work determines the complexity status of two problems related to finding the smallest number k such that a given graph is a partial k-tree and presents an algorithm with polynomially bounded (but exponential in k) worst case time complexity.
Abstract: A k-tree is a graph that can be reduced to the k-complete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a k-tree. This problem is motivated by the existence of polynomial time algorithms for many combinatorial problems on graphs when the graph is constrained to be a partial k-tree for fixed k. These algorithms have practical applications in areas such as reliability, concurrent broadcasting and evaluation of queries in a relational database system. We determine the complexity status of two problems related to finding the smallest number k such that a given graph is a partial k-tree. First, the corresponding decision problem is NP-complete. Second, for a fixed (predetermined) value of k, we present an algorithm with polynomially bounded (but exponential in k) worst case time complexity. Previously, this problem had only been solved for $k = 1,2,3$.
1,350 citations