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Subexponential parameterized algorithms on graphs of boundedgenus and Hminorfree Graphs
"... ... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossing ..."
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Cited by 62 (21 self)
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... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossingminorfree graphs, and anyclass of graphs that is closed under taking minors. Specifically, the running time is 2O(pk)nh, where h is a constant depending onlyon H, which is polynomial for k = O(log² n). We introducea general approach for developing algorithms on Hminorfreegraphs, based on structural results about Hminorfree graphs at the
Parameterized complexity and approximation algorithms
 Comput. J
, 2006
"... Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We ..."
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Cited by 58 (2 self)
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Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research. 1.
Efficient Approximation Schemes for Geometric Problems?⋆
"... Abstract. An EPTAS (efficient PTAS) is an approximation scheme where ǫ does not appear in the exponent of n, i.e., the running time is f(ǫ) ·nc. We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answe ..."
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Abstract. An EPTAS (efficient PTAS) is an approximation scheme where ǫ does not appear in the exponent of n, i.e., the running time is f(ǫ) ·nc. We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answering an open question of Alber and Fiala [2], we show that Maximum Independent Set is W[1]complete for the intersection graphs of unit disks and axisparallel unit squares in the plane. A standard consequence of this result is that the nO(1/ǫ) time PTAS of Hunt et al. [11] for Maximum Independent Set on unit disk graphs cannot be improved to an EPTAS. Similar results are obtained for the problem of covering points with squares. 1
Subexponential Parameterized Algorithms on BoundedGenus Graphs and HMinorFree Graphs
"... cfl 202005 ACM 00045411/202005/0100100001 $5.00 ..."
On Two Techniques of Combining Branching and Treewidth
, 2006
"... Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations o ..."
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Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations of these techniques. The idea behind these algorithms is very natural: If a parameter like the treewidth of a graph is small, algorithms based on dynamic programming perform well. On the other side, if the treewidth is large, then there must be vertices of high degree in the graph, which is good for branching algorithms. We give several examples of possible combinations of branching and programming which provide the fastest known algorithms for a number of NP hard problems. All our algorithms require nontrivial balancing of these two techniques. In the first approach the algorithm either performs fast branching, or if there is an obstacle for fast branching, this obstacle is used for the construction of a path decomposition of small width for the original graph. Using this approach we give the fastest known algorithms for Minimum Maximal Matching and for counting all 3colorings of a graph. In the second approach the branching occurs until the algorithm reaches a subproblem with a small number of edges (and here the right choice of the size of subproblems is crucial) and then dynamic programming is applied on these subproblems of small width. We exemplify this approach by giving the fastest known algorithm to count all minimum weighted dominating sets of a graph. We also discuss how similar techniques can be used to design faster parameterized algorithms.
Exact and Parameterized Algorithms for Max Internal Spanning Tree
, 2009
"... We consider the NPhard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamicprogramming algorithms for general and degreebounded graphs have running times of the form O ∗ (c n) with c ≤ 2 ..."
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We consider the NPhard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamicprogramming algorithms for general and degreebounded graphs have running times of the form O ∗ (c n) with c ≤ 2. For graphs with bounded degree, c < 2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8669 n) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364 k n O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.