## Fixed parameter algorithms for planar dominating set and related problems (2000)

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### BibTeX

@INPROCEEDINGS{Alber00fixedparameter,

author = {Jochen Alber and Hans L. Bodlaender and Henning Fernau and Rolf Niedermeier},

title = {Fixed parameter algorithms for planar dominating set and related problems},

booktitle = {},

year = {2000},

pages = {97--110},

publisher = {Springer}

}

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### Abstract

We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ kn), where c = 36√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 √ k in O(c1 n + n2) time, where c1 = 236√34 and 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. Keywords. NP-complete problems, fixed parameter tractability, planar graphs, planar dominating set, face cover, outerplanarity, treewidth.

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Citation Context ..., given a graph G = (V, E) and a positive integer k, whether or not there exists a k-dominating set, is among the core problems in algorithms, combinatorial optimization, and computational complexity =-=[4, 16, 27, 32, 44]-=-. The problem is NP-complete, even when restricted to planar graphs with maximum vertex degree 3 and to planar graphs that are regular of degree 4 [27]. Coping with NP-hard problems. Despite their int... |

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Citation Context ..., given a graph G = (V, E) and a positive integer k, whether or not there exists a k-dominating set, is among the core problems in algorithms, combinatorial optimization, and computational complexity =-=[4, 16, 27, 32, 44]-=-. The problem is NP-complete, even when restricted to planar graphs with maximum vertex degree 3 and to planar graphs that are regular of degree 4 [27]. Coping with NP-hard problems. Despite their int... |

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Citation Context ..., given a graph G = (V, E) and a positive integer k, whether or not there exists a k-dominating set, is among the core problems in algorithms, combinatorial optimization, and computational complexity =-=[4, 16, 27, 32, 44]-=-. The problem is NP-complete, even when restricted to planar graphs with maximum vertex degree 3 and to planar graphs that are regular of degree 4 [27]. Coping with NP-hard problems. Despite their int... |

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Citation Context ...2), the set B(C) as given in Lemma 6 is called the boundary cycle of C. 2.2 Domination and treewidth The main tool we use in our algorithm is the concept of tree decompositions as, e.g., described in =-=[9]-=-. Definition 8 Let G = (V, E) be a graph. A tree decomposition of G is a pair #{X i | i # I}, T #, where each X i is a subset of V , called a bag, and T is a tree with the elements of I as nodes. The ... |

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Citation Context ...iminaries: Domination, r-outerplanarity, and treewidth In this section, we provide necessary notions and some known results. We assume familiarity with basic graph-theoretical notation as provided in =-=[19, 38]-=-. In particular, for a graph G = (V, E) and a subset V # # V , the subgraph induced by the vertices V \ V # will frequently be denoted by G - V # . If V # = {v} is a singleton, we write G- v instead o... |

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Citation Context ...n be derived from this observation and the fact that an routerplanar graph has treewidth of at most 3r - 1 (as exhibited in Section 2). Alternatively, the general logical framework of Frick and Grohe =-=[26]-=- easily proves the fixed parameter tractability of planar k-dominating set. Downey and Fellows [21, 22] give an O(11 k n) time algorithm, the so far best known time bound for planar k-dominating set. ... |

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Citation Context ...et is an interesting and promising research question. Fixed parameter tractability. Lately, it has become popular to cope with computational intractability in a di#erent way: parameterized complexity =-=[1, 22, 23, 24]. Here, th-=-e basic observation is that, for many hard problems, the seemingly inherent combinatorial explosion can be restricted to a "small part" of the input, the parameter. For instance, the k-verte... |

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Citation Context ... 3r - 1 (as exhibited in Section 2). Alternatively, the general logical framework of Frick and Grohe [26] easily proves the fixed parameter tractability of planar k-dominating set. Downey and Fellows =-=[21, 22]-=- give an O(11 k n) time algorithm, the so far best known time bound for planar k-dominating set. Relevance of (planar) dominating set. The literature on dominating set problems in mathematics, compute... |

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Citation Context ...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 2 proven to be complete for W [2] (see =-=[20]), a "-=-;complexity class of parameterized intractability" (refer to Downey and Fellows [22] for details). However, planar k-dominating set is fixed parameter tractable. This already easily follows from ... |

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Citation Context ... in time 3 # N , where N is the number of nodes of the tree decomposition. 7 Proof. The theorem can be proven by using dynamic programming techniques, as described in a more general context, e.g., in =-=[9, 48]-=-. For the sake of preciseness, we outline how these techniques apply to solving dominating set in the claimed running time. Let X = #{X i | i # I}, T # be a tree decomposition for the graph G = (V, E)... |

44 | A general method to speed up fixedparameter-tractable algorithms
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Citation Context ...ady guarantees fixed parameter tractability for this problem. 31 small problem kernel, could be of high practical relevance, in particular in combination with the interleaving techniques presented in =-=[42]-=-. Acknowledgements. We are indebted to Ton Kloks for starting this research with us and for many discussions on the topic of this paper. We thank Peter Rossmanith for discussions on di#erent approache... |

43 | Upper bounds for Vertex Cover further improved
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Citation Context ...herent combinatorial explosion can be restricted to a "small part" of the input, the parameter. For instance, the k-vertex cover problem can be solved by an algorithm with running time O(kn =-=+ 1.3 k ) [14, 41]-=-, where the parameter k is a bound on the maximum size of the vertex cover set we are looking for and n is the number of vertices in the given graph. The fundamental assumption is k # n. As can easily... |

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Citation Context ...iminaries: Domination, r-outerplanarity, and treewidth In this section, we provide necessary notions and some known results. We assume familiarity with basic graph-theoretical notation as provided in =-=[19, 38]-=-. In particular, for a graph G = (V, E) and a subset V # # V , the subgraph induced by the vertices V \ V # will frequently be denoted by G - V # . If V # = {v} is a singleton, we write G- v instead o... |

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Citation Context ...e, [29, 30, 31, 37, 46]. In particular, note that many papers have been published on domination problems for special graph classes [11] and/or variations of the fundamental problem, see, for example, =-=[2, 6, 12, 13, 15, 25, 17, 34, 35, 40, 49]-=-. Our main result. We present fixed parameter tractability results for planar k-dominating set and related problems. Our main result is to prove a new and perhaps surprising structural relationship: W... |

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Citation Context ...unately, 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 2 proven to be complete for W =-=[2] (see [20]-=-), a "complexity class of parameterized intractability" (refer to Downey and Fellows [22] for details). However, planar k-dominating set is fixed parameter tractable. This already easily fol... |

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Citation Context ...e the exponent of the exponential term is growing sublinearly. 3 Further contributions of our work. Our new method can also be used to significantly improve a known bound for the k-face cover problem =-=[7, 22, 45]-=-. The problem is defined as follows [22, 7, 45]: Given a plane graph G, i.e., a graph with a fixed embedding in the plane and a positive integer k, is there a set of at most k faces (also called disks... |

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Citation Context ...et is an interesting and promising research question. Fixed parameter tractability. Lately, it has become popular to cope with computational intractability in a di#erent way: parameterized complexity =-=[1, 22, 23, 24]. Here, th-=-e basic observation is that, for many hard problems, the seemingly inherent combinatorial explosion can be restricted to a "small part" of the input, the parameter. For instance, the k-verte... |

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Citation Context ...set. The literature on dominating set problems in mathematics, computer science, and their applications (e.g., computational biology) is huge. Hence, we only refer to some surveys here, for instance, =-=[29, 30, 31, 37, 46]-=-. In particular, note that many papers have been published on domination problems for special graph classes [11] and/or variations of the fundamental problem, see, for example, [2, 6, 12, 13, 15, 25, ... |

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Citation Context ...h G = (V, E), a property P , and a positive integer k, whether or not there exists a k-dominating set with property P . Examples for such problems are: . the k-independent dominating set problem, see =-=[47, 48]-=- or [22, p.464], where the property P (D) of the k-dominating set D is that D is independent, . the k-total dominating set problem, see [47, 48], where the property P (D) of the k-dominating set D is ... |

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Citation Context ...perty P (D) of the k-dominating set D is that each vertex which is not in D has exactly one neighbor in D, . the k-perfect independent dominating set problem, also known as the k-perfect code problem =-=[22, 28, 47, 48]-=-, where the k-dominating set has to be perfect and independent, and . the k-total perfect dominating set problem [47, 48], where the kdominating set has to be total and perfect. For all these instance... |

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Citation Context ...earlier in the algorithm. Also, note that, when bookkeeping how the minima in Step 2 were obtained, this algorithm constructs a dominating set D corresponding to #(G). # We remark that Aspvall et al. =-=[3]-=- addressed the memory requirement problem arising in the type of algorithms described above. They suggested a method to minimize the sum of the sizes of the tables that need to be stored simultaneousl... |

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Citation Context ...e, [29, 30, 31, 37, 46]. In particular, note that many papers have been published on domination problems for special graph classes [11] and/or variations of the fundamental problem, see, for example, =-=[2, 6, 12, 13, 15, 25, 17, 34, 35, 40, 49]-=-. Our main result. We present fixed parameter tractability results for planar k-dominating set and related problems. Our main result is to prove a new and perhaps surprising structural relationship: W... |

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Citation Context ...set. The literature on dominating set problems in mathematics, computer science, and their applications (e.g., computational biology) is huge. Hence, we only refer to some surveys here, for instance, =-=[29, 30, 31, 37, 46]-=-. In particular, note that many papers have been published on domination problems for special graph classes [11] and/or variations of the fundamental problem, see, for example, [2, 6, 12, 13, 15, 25, ... |

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Citation Context ...e, [29, 30, 31, 37, 46]. In particular, note that many papers have been published on domination problems for special graph classes [11] and/or variations of the fundamental problem, see, for example, =-=[2, 6, 12, 13, 15, 25, 17, 34, 35, 40, 49]-=-. Our main result. We present fixed parameter tractability results for planar k-dominating set and related problems. Our main result is to prove a new and perhaps surprising structural relationship: W... |

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Citation Context ...et is an interesting and promising research question. Fixed parameter tractability. Lately, it has become popular to cope with computational intractability in a di#erent way: parameterized complexity =-=[1, 22, 23, 24]. Here, th-=-e basic observation is that, for many hard problems, the seemingly inherent combinatorial explosion can be restricted to a "small part" of the input, the parameter. For instance, the k-verte... |

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Citation Context ... and L i+2 directly, and use that set instead of S i as defined in the proof of Section 3.2. Such a minimum size separator can be computed with well known techniques based on maximum flow (see, e.g., =-=[33]-=-). Our algorithm proceeds in the following steps: 1. Embed the planar graph G = (V, E) crossing-free into the plane. Determine the outerplanarity number r of this embedding and all layers L 1 , . . . ... |

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Citation Context ...e the exponent of the exponential term is growing sublinearly. 3 Further contributions of our work. Our new method can also be used to significantly improve a known bound for the k-face cover problem =-=[7, 22, 45]-=-. The problem is defined as follows [22, 7, 45]: Given a plane graph G, i.e., a graph with a fixed embedding in the plane and a positive integer k, is there a set of at most k faces (also called disks... |

3 | A general method to speed up algorithms - Niedermeier, Rossmanith - 2000 |

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Citation Context ...set. The literature on dominating set problems in mathematics, computer science, and their applications (e.g., computational biology) is huge. Hence, we only refer to some surveys here, for instance, =-=[29, 30, 31, 37, 46]-=-. In particular, note that many papers have been published on domination problems for special graph classes [11] and/or variations of the fundamental problem, see, for example, [2, 6, 12, 13, 15, 25, ... |

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