Parameterized Complexity: Exponential Speed-Up for Planar Graph Problems

Alber, Jochen; Fernau, Henning; Niedermeier, Rolf

doi:10.1007/3-540-48224-5_22

Cited by 55 publications

(88 citation statements)

References 15 publications

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“…The new entry for a specific coloring c (i+1) ∈ {1, 0 outer , 0 inner } |S i+1 | in table A i+1 can be obtained from minimizing over all colorings c (i) ∈ {1, 0 outer , 0 inner } |S i | in mapping A i and applying Baker's algorithm to the graph G i precolored according to c (i) and c (i+1) . Working out the details of this approach yields a running time of O(c √ k n) for some slightly better constant c than the one obtained in Theorem 31 (see [2]). However, we want to point out that the algorithm claimed by Baker is not worked out in detail for the DOMINATING SET problem, and, hence, it is hard to verify whether it can deal with precolored graphs, a basic assumption we need for this approach.…”

Section: The Algorithmmentioning

confidence: 90%

“…Indeed, this seems to be the first non-trivial result for an NP-hard, fixed parameter tractable problem where the exponent of the exponential term is growing sublinearly. Very recently, on the one hand, the basic concepts of this paper could be generalized to a wider spectrum of planar graph problems [2] (also see [3] for an approach based on planar separator theorems instead of tree decompositions). On the other hand, Cai et al [16] showed that our result is in a way optimal: They state that there is no time 2 o( √ k) n O (1) algorithm unless 3SAT ∈ DTIME (2 o(n) ), which is considered to be unlikely.…”

mentioning

confidence: 98%

“…The fundamental assumption is k n. As can be easily seen, this yields an efficient, practical algorithm for small values of k. In general, a problem is called fixed parameter tractable if it can be solved in time f (k)n O (1) for an arbitrary function f which depends only on k. Unfortunately, according to the theory of parameterized complexity, it is very unlikely that the k-DOMINATING SET problem is fixed parameter tractable. On the contrary, it was proven to be complete for W [2] (see [21]), a "complexity class of parameterized intractability" (refer to [23] for details). However, k-DOMINATING SET on planar graphs is fixed parameter tractable.…”

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confidence: 99%

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Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Bodlaender²,

et al. 2002

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We present an algorithm that constructively produces a solution to the k-DOMINATING SET problem for planar graphs in time O(c √ k n), where c = 4 6 √ 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O( √ γ (G)), and that such a tree decomposition can be found in O( √ γ (G)n) time. The same technique can be used to show that the k-FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c √ k 1 n) time, where c 1 = 3 36 √ 34and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-DOMINATING SET, e.g., k-INDEPENDENT DOMINATING SET and k-WEIGHTED DOMINATING SET.

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Section: The Algorithmmentioning

confidence: 90%

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confidence: 98%

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confidence: 99%

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Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Bodlaender²,

et al. 2002

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“…Therefore, at least 80r − 40r of the 80r paths in Y encountered on P 1 [i, i + 1] do not leave the ith interval. This proves(2).…”

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confidence: 67%

“…2 The more general connection for H -minor-free graphs has been obtained recently: Fig. 1 The (4 × 5)-grid and the (8 × 5)-wall Theorem 2 ([15]) For any fixed graph H , every H -minor-free graph of treewidth w has an (w + 1) × (w + 1) grid graph as a minor.…”

Section: Grid Minorsmentioning

confidence: 99%

Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction

Demaine¹,

Hajiaghayi

Kawarabayashi

2007

Algorithmica

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We explore three important avenues of research in algorithmic graphminor theory, which all stem from a key min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner's Conjecture about the structure of minor-closed graph properties.First, we obtain the only known polynomial min-max relation for graphs that do not exclude any fixed minor, namely, map graphs and power graphs. Second, we obtain explicit (and improved) bounds on the min-max relation for an important class of graphs excluding a minor, namely, K 3,k -minor-free graphs, using new techniques that do not rely on Graph Minor Theory. These two avenues lead to faster fixedparameter algorithms for two families of graph problems, called minor-bidimensional and contraction-bidimensional parameters, which include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour. Third, we disprove a variation of Wagner's Conjecture for the case of graph contractions in general graphs, and in a sense characterize which graphs satisfy the Algorithmica (2009) 54: 142-180 143 variation. This result demonstrates the limitations of a general theory of algorithms for the family of contraction-closed problems (which includes, for example, the celebrated dominating-set problem). If this conjecture had been true, we would have had an extremely powerful tool for proving the existence of efficient algorithms for any contraction-closed problem, like we do for minor-closed problems via Graph Minor Theory.

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New upper bounds on the decomposability of planar graphs

Fomin

Thilikos

2005

Journal of Graph Theory

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DOI 10.1002/jgt.20121 Abstract: It is known that a planar graph on n vertices has branch-width/ tree-width bounded by ffiffiffi n p. In many algorithmic applications, it is useful to have a small bound on the constant . We give a proof of the best, so far, upper bound for the constant . In particular, for the case of tree-width, < 3:182 and for the case of branch-width, < 2:122. Our proof is based on the planar separation theorem of Alon, Seymour, and Thomas and some min–max theorems of Robertson and Seymour from the graph minors series. We also discuss some algorithmic consequences of this result

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Parameterized Complexity: Exponential Speed-Up for Planar Graph Problems

Cited by 55 publications

References 15 publications

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction

New upper bounds on the decomposability of planar graphs

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