An algebraic theory of graph reduction

Arnborg, Stefan; Courcelle, Bruno; Proskurowski, Andrzej; Seese, Detlef

doi:10.1007/bfb0017382

Cited by 28 publications

(43 citation statements)

References 29 publications

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“…In addition, treewidth has proven very interesting from a graph-theoretic perspective, one of its most important attributes being that it can be equivalently defined in many seemingly unrelated ways. For example treewidth is connected to chordal graphs, elimination schemes, partial k-trees, cops-and-robber games [2,3], reduction rules [4] and brambles [2]. Thus, treewidth has proven so algorithmically successful and graph-theoretically robust that it is widely considered the ''right'' complexity measure for undirected graphs.…”

Section: Introductionmentioning

confidence: 99%

On the algorithmic effectiveness of digraph decompositions and complexity measures

Lampis¹,

Kaouri²,

Mitsou³

2011

Discrete Optimization

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a b s t r a c tWe place our focus on the gap between treewidth's success in producing fixed-parameter polynomial algorithms for hard graph problems, and specifically Hamiltonian Circuit and Max Cut, and the failure of its directed variants (directed treewidth (Johnson et al., 2001 [13]), DAG-width (Obdrzálek, 2006 [14]) and Kelly-width (Hunter and Kreutzer, 2007 [15]) to replicate it in the realm of digraphs. We answer the question of why this gap exists by giving two hardness results: we show that Directed Hamiltonian Circuit is W [2]-hard when the parameter is the width of the input graph, for any of these widths, and that Max Di Cut remains NP-hard even when restricted to DAGs, which have the minimum possible width under all these definitions. Along the way, we extend our reduction for Directed Hamiltonian Circuit to show that the related Minimum Leaf Outbranching problem is also W [2]-hard when naturally parameterized by the number of leaves of the solution, even if the input graph has constant width. All our results also apply to directed pathwidth and cycle rank.

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Section: Introductionmentioning

confidence: 99%

On the algorithmic effectiveness of digraph decompositions and complexity measures

Lampis¹,

Kaouri²,

Mitsou³

2011

Discrete Optimization

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“…In [7,8,10], algorithms are presented that solve a number of NP-hard problems in linear time if the given graph has bounded treewidth and a tree decomposition of the graph is given as part of the input. An interesting framework for solving these problems on graphs of bounded treewidth without computing a tree decomposition of the graph is presented in [9]; the resulting algorithms are based on graph reduction and take linear time, but use superlinear space.…”

Section: Previous Workmentioning

confidence: 99%

I/O-Efficient Algorithms for Graphs of Bounded Treewidth

Maheshwari

Zeh

2007

Algorithmica

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We present an algorithm that takes O(sort(N )) I/Os (sort(N ) = ((N/ (DB)) log M/B (N/B)) is the number of I/Os it takes to sort N data items) to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N , where k is a constant. 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 single-source shortest path problem and a number of problems that are NP-hard 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 depth-first 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 k-tree 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/O-efficient 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/O-efficient algorithm for finding a maximal inAn abstract of this paper was presented at

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“…for every pair of nonnegative integers j 1 and j 2 with j 1 + j 2 ; and (b) there is an active count γ L on X L such that…”

Section: Algorithmmentioning

confidence: 99%

“…graph with tree-width bounded by a fixed constant k. The class of partial k-trees is fairly large, and includes trees, outerplanar graphs, series-parallel graphs, etc. It is known that many combinatorial problems can be solved very efficiently for partial k-trees even if the problems are NP-hard for general graphs [2,3,[8][9][10]. Such classes of problems have been characterized in terms of "forbidden subgraphs" or "extended monadic second-order logic" [2,3,[8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Algorithms for finding distance-edge-colorings of graphs

Ito

Kato

Zhou

et al. 2007

Journal of Discrete Algorithms

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For a bounded integer , we wish to color all edges of a graph G so that any two edges within distance have different colors. Such a coloring is called a distance-edge-coloring or an -edge-coloring of G. The distance-edge-coloring problem is to compute the minimum number of colors required for a distance-edge-coloring of a given graph G. A partial k-tree is a graph with tree-width bounded by a fixed constant k. We first present a polynomial-time exact algorithm to solve the problem for partial k-trees, and then give a polynomial-time 2-approximation algorithm for planar graphs.

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An algebraic theory of graph reduction

Cited by 28 publications

References 29 publications

On the algorithmic effectiveness of digraph decompositions and complexity measures

On the algorithmic effectiveness of digraph decompositions and complexity measures

I/O-Efficient Algorithms for Graphs of Bounded Treewidth

Algorithms for finding distance-edge-colorings of graphs

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