On the existence of subexponential parameterized algorithms

Cai, Liming; Juedes, David

doi:10.1016/s0022-0000(03)00074-6

Cited by 123 publications

(90 citation statements)

References 20 publications

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“…Notice that the above result is, in a sense, optimal, as, due to the complexity bounds in [18], a 2 O( √ k) · n O(1) -step parameterized algorithm is the best we may expect for several bidimensional parameters, even on planar graphs. The metaalgorithmic machinery that we employed above in order to prove Theorem 7 is known as Bidimensionality Theory and was introduced for the first time in [27], while some preliminary ideas had already appeared in [4,52].…”

Section: Theorem 7 Let H Be An R-vertex Graph and Let P Be A Graph Pmentioning

confidence: 84%

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 7 Let H Be An R-vertex Graph and Let P Be A Graph Pmentioning

confidence: 84%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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“…It would be interesting to investigate the "optimality" of the form of our FPT results in the sense of [CJ03,DEFPR03]. Can it be shown that there is no O(2…”

Section: Winning the Fpt Runtime Racementioning

confidence: 99%

Finding k Disjoint Triangles in an Arbitrary Graph

Fellows

Heggernes

Rosamond

et al. 2004

Graph-Theoretic Concepts in Computer Science

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We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. For H = K 3 (the triangle) we give an O(2 2k log k+1.869k n 2 ) algorithm, and for general H an O(2 k|H| log k+2k|H| log |H| n |H| ) algorithm. We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph. For H = K 3 this leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k 3 ) vertices in polynomial time. INTRODUCTIONFor a fixed graph H and an input graph G, the H-packing problem asks for the maximum number of vertex-disjoint copies of H in G. The K 2 -packing (edge packing) problem, which is equivalent to maximum matching, played a central role in the history of classical computational complexity. The first step towards the dichotomy of 'good' (polynomialtime) versus 'presumably-not-good' (NP-hard) was made in a paper on maximum matching from 1965 [E65], which gave a polynomial time algorithm for that problem. On the other hand, the K 3 -packing (triangle packing) problem, which is our main concern in this paper, is NP-hard [HK78].Recently, there has been a growing interest in the area of exact exponential-time algorithms for NP-hard problems. When measuring time in the classical way, simply by the 1

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“…While there is a strong evidence that most fixed-parameter algorithms cannot have running times 2 o(k) · n O(1) (see [13,35,43]), for planar graphs it is possible to design subexponential parameterized algorithms with running times of the type 2 O( √ k) · n O(1) (see [13,15] for further lower bounds on planar graphs). For example, Planar k-Vertex Cover can be solved in O(2 3.57 [27].…”

Section: Introductionmentioning

confidence: 99%

Subexponential Parameterized Algorithms

Fomin¹

2015

Encyclopedia of Algorithms

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We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch-(or tree-) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of sub-exponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2 O( √ k) · n O(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter.

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On the existence of subexponential parameterized algorithms

Cited by 123 publications

References 20 publications

Graph Minors and Parameterized Algorithm Design

Graph Minors and Parameterized Algorithm Design

Finding k Disjoint Triangles in an Arbitrary Graph

Subexponential Parameterized Algorithms

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