Results 1  10
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21
Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs
, 1993
"... In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most ..."
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Cited by 49 (11 self)
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In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most k, and that use O(V) time. In contrast with previous solutions, our algorithms do not rely on nonconstructive reasoning, and are single exponential in k and l. This result can be combined with a result of Reed [37], yielding explicit O(n log n) algorithms for the problem, given a graph G, to determine whether the treewidth (or pathwidth) of G is at most k, and if so, to find a tree (or path)decomposition of width at most k (k constant). Also, Bodlaender [13] has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constants k, there exists a polynomial time algorithm, that, when given a graph G = (V; E) with treewidth k, computes the pathwidth of G and a minimum path decomposition of G.
Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms
, 1995
"... We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a consta ..."
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Cited by 35 (4 self)
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We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(ff(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(n fi ), for any constant 0 ! fi ! 1. Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time.
Constructive Linear Time Algorithms for Branchwidth
, 1997
"... We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The noti ..."
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Cited by 27 (6 self)
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We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The notion of branchwidth has a close relationship to the more wellknown notion of treewidth, a notion that has come to play a large role in many recent investigations in algorithmic graph theory. (See Section 2 for definitions of treewidth and branchwidth.) One reason for the interest in this notion is that many graph problems can be solved by linear time algorithms, when the inputs are restricted to graphs with some uniform upper bound on their treewidth. Most of these algorithms first try to find a tree decomposition of small width, and then utilize the advantages of the tree structure of the decomposition (see [1], [4]). The branchwidth of a graph differs from its treewidth by at most a multipl...
I/Oefficient algorithms for graphs of bounded treewidth
 In Proceedings of the 12th Annual ACMSIAM Symposium on Discrete Algorithms (SODA’2001
, 2001
"... We present an algorithm that takes O(sort(N)) I/Os 1 to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in O(N/(DB)) I/Os, including th ..."
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Cited by 15 (5 self)
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We present an algorithm that takes O(sort(N)) I/Os 1 to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in O(N/(DB)) I/Os, including the singlesource shortest path problem and a number of problems that are NPhard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depthfirst search in G in O(N/(DB)) I/Os. As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a ktree in O(sort(N)) I/Os. The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/Oefficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/Oefficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take O(sort(V +E)) I/Os. The maximal matching algorithm is used in the tree decomposition algorithm.
Graph operations characterizing rankwidth and balanced graph expressions
, 2007
"... Graph complexity measures like treewidth, cliquewidth, NLCwidth and rankwidth are important because they yield Fixed Parameter Tractable algorithms. Rankwidth is based on ranks of adjacency matrices of graphs over GF(2). We propose here algebraic operations on graphs that characterize rankwidt ..."
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Cited by 7 (2 self)
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Graph complexity measures like treewidth, cliquewidth, NLCwidth and rankwidth are important because they yield Fixed Parameter Tractable algorithms. Rankwidth is based on ranks of adjacency matrices of graphs over GF(2). We propose here algebraic operations on graphs that characterize rankwidth. For algorithmic purposes, it is important to represent graphs by balanced terms. We give a unique theorem that generalizes several “balancing theorems” for treewidth and cliquewidth. New results are obtained for rankwidth and a variant of cliquewidth, called mcliquewidth.
Reduction Algorithms for Graphs with Small Treewidth
 In Proceedings 19th International Workshop on GraphTheoretic Concepts in Computer Science WG'93
, 1995
"... This paper presents some new ideas and results on graph reduction applied to graphs with bounded treewidth. Arnborg et al. [2] have shown that many decision problems on graphs can be solved in linear time on graphs with bounded treewidth, by using a finite set of reduction rules. We show that this m ..."
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Cited by 6 (5 self)
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This paper presents some new ideas and results on graph reduction applied to graphs with bounded treewidth. Arnborg et al. [2] have shown that many decision problems on graphs can be solved in linear time on graphs with bounded treewidth, by using a finite set of reduction rules. We show that this method can also be used to solve the construction variants of many of these problems, and to solve a number of optimization problems, and to solve construction variants of many of these optimization problems. For example, the construction variants of decision problems that are definable in monadic second order logic can be solved in this way. Examples of optimization problems that can be solved in this way are INDEPENDENT SET, INDUCED BOUNDED DEGREE SUBGRAPH, PARTITION INTO CLIQUES and HAMILTONIAN COMPLETION NUMBER. We also show that the results of [6] can be applied to these reduction algorithms, which results in parallel algorithms that use O(n) operations and O(log n log n) time on an ...
Parallel Algorithms for Series Parallel Graphs
 Algorithmica
, 1996
"... In this paper, a parallel algorithm is given that, given a graph G = (V; E), decides whether G is a series parallel graph, and if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log E log E) time and O(E) operations on an EREW PRAM, and O(lo ..."
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Cited by 6 (4 self)
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In this paper, a parallel algorithm is given that, given a graph G = (V; E), decides whether G is a series parallel graph, and if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log E log E) time and O(E) operations on an EREW PRAM, and O(log E) time and O(E) operations on a CRCW PRAM (note that if G is a simple series parallel graph, then E = O(V)). With the same time and processor resources, a treedecomposition of width at most two can be built of a given series parallel graph, and hence, very efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs, including many well known problems, e.g., all problems that can be stated in monadic second order logic. The results hold for undirected series parallel graphs graphs, as well as for directed series parallel graphs.
The maximum subforest problem: Approximation and exact algorithms
 IN PROC. 9TH SYMPOSIUM ON DISCRETE ALGORITHMS (SODA 98)
, 1998
"... We study the maximum subforest problem: Given a tree G and a set of trees H, find a subgraph G ′ of G such that G ′ does not contain a subtree isomorphic to a tree from H, and the number of edges in G ′ is maximum. We give a polynomial time approximation scheme for this problem. We also give an exac ..."
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Cited by 6 (3 self)
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We study the maximum subforest problem: Given a tree G and a set of trees H, find a subgraph G ′ of G such that G ′ does not contain a subtree isomorphic to a tree from H, and the number of edges in G ′ is maximum. We give a polynomial time approximation scheme for this problem. We also give an exact algorithm for this problem whose time complexity is 2 O(k2 / log k) n, where n is the number of vertices in G, and k is the total number of vertices in H.