Graph Minors .XIII. The Disjoint Paths Problem

Robertson, Neil; Seymour, Paul

doi:10.1006/jctb.1995.1006

Cited by 1,039 publications

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“…Actually, Robertson and Seymour in [108] describe an O k (n 3 )-step algorithm that solves a generalization of the H-Minor Checking and another celebrated problem, namely the k-Disjoint Paths problem. In Section 5, we give a rough description of the main ingredients of the algorithm in Theorem 2 especially for the k-Disjoint Paths problem.…”

Section: Theorem 2 (Robertson and Seymour[108]) One Can Construct Anmentioning

confidence: 99%

“…The bad news is that, according to the algorithm in [108] and the proof of its correctness in [111] and [107], the values of this function are immense 4 involving r 2's. Clearly, such type of constants may create reasonable doubts to computer scientists on whether such an algorithm may be considered to be an "algorithm" of some practical meaning.…”

Section: Theorem 2 (Robertson and Seymour[108]) One Can Construct Anmentioning

confidence: 99%

“…The algorithm in Lemma 1 works as follows: First of all, it uses an FPTapproximation algorithm for treewidth, i.e., an algorithm that given a graph G and an integer q, either outputs a tree decomposition of G of width at most α · q or reports that tw(G) > q in z(q) · n β steps. Various algorithms of this type have been proposed in [7,14,88,100,108] for different trade-offs between z, α, and β. Among them, we pick the one form [7] where z(q) = 2 4.38·q , α = 4, and β = 2.…”

Section: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 99%

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Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique.

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Section: Theorem 2 (Robertson and Seymour[108]) One Can Construct Anmentioning

confidence: 99%

Section: Theorem 2 (Robertson and Seymour[108]) One Can Construct Anmentioning

confidence: 99%

Section: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 99%

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Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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“…Observe that, in a general (di)graph, the problem of deciding whether there exist edge-disjoint paths between given pairs of vertices is NP-complete [Kar75] (even if the graph is a square grid [Kv82]). When the number of pairs of vertices is bounded by a constant, the disjoint paths problem is polynomial in undirected graphs [RS95], NP-complete in directed graphs [LR80] (even with only two pairs of vertices [FHW80]), and polynomial in symmetric directed graphs [JP09].…”

Section: Background and Motivationmentioning

confidence: 99%

Edge-Simple Circuits through 10 Ordered Vertices in Square Grids

Coudert

Giroire

2009

Lecture Notes in Computer Science

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A circuit in a simple undirected graph G = (V, E) is a sequence of vertices {v 1 , v 2 , . . . , v k+1 } such that v 1 = v k+1 and {v i , v i+i } ∈ E for i = 1, . . . , k. A circuit C is said to be edge-simple if no edge of G is used twice in C. In this article we study the following problem: which is the largest integer k such that, given any subset of k ordered vertices of an infinite square grid, there exists an edge-simple circuit visiting the k vertices in the prescribed order? We prove that k = 10. To this end, we first provide a counterexample implying that k < 11. To show that k ≥ 10, we introduce a methodology, based on the notion of core graph, to reduce drastically the number of possible vertex configurations, and then we test each one of the resulting configurations with an ILP solver.

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“…Once an embedding in a k-tree is given for a graph, many combinatorial problems can be solved in time linear in the size of the graph [2-4, 6, 9, 11, 17, 18, 19, 35] (the constant of proportionality depends however on the value of k). An O(nz ) approximate embedding algorithm was developed by Robertson and Seymour [30,31]. Various improvements are possible (see e.g., Courcelle [17], Lagergren [25], and Bodlaender [12,13].…”

Section: Introductionmentioning

confidence: 99%

An algebraic theory of graph reduction

Arnborg

Courcelle

Proskurowski

et al.

Lecture Notes in Computer Science

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Abstract, Wc show how membership in classes of graphs definable m monwhc second-order ]oglc and of bounded treewldth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that wc describe an algorlthm that wdl produce, from J formula in monxhc second-order Ioglc and an mleger k such that the class dcfmed by the formul~IS of treewidth s k, a set of rewrite rules that rcducxs any member of the elms to one of' firrltely many graphs, in a number of steps bounded by the size c~f the graph. This rcductmn syjtem ymlds an algorlthm that runs m time linear m the size of the graph. We illustrate our results with rcductlon systems that recognux some families of outerplanar and planar graphs.

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Graph Minors .XIII. The Disjoint Paths Problem

Cited by 1,039 publications

References 0 publications

Graph Minors and Parameterized Algorithm Design

Graph Minors and Parameterized Algorithm Design

Edge-Simple Circuits through 10 Ordered Vertices in Square Grids

An algebraic theory of graph reduction

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