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41
Parameterized Complexity: Exponential SpeedUp for Planar Graph Problems
 in Electronic Colloquium on Computational Complexity (ECCC
, 2001
"... A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniqu ..."
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Cited by 61 (21 self)
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A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.
Graph separators: a parameterized view
 Journal of Computer and System Sciences
, 2001
"... Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. ..."
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Cited by 30 (14 self)
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Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p
The bchromatic number of a graph
 Discrete Applied Math
, 1999
"... The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V1, V2,..., Vk into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all ..."
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Cited by 21 (0 self)
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The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V1, V2,..., Vk into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G. We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the bchromatic number, ϕ(G), of G. We prove that determining ϕ(G) is NPhard for general graphs, but polynomialtime solvable for trees.
Randomised Techniques in Combinatorial Algorithmics
, 1999
"... ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ..."
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Cited by 20 (7 self)
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ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...
The stable roommates problem with ties
 J. Algorithms
, 2002
"... We study the variant of the wellknown Stable Roommates problem in which participants are permitted to express ties in their preference lists. In this setting, more than one definition of stability is possible. Here we consider two of these stability criteria, socalled superstability and weak stab ..."
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Cited by 17 (3 self)
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We study the variant of the wellknown Stable Roommates problem in which participants are permitted to express ties in their preference lists. In this setting, more than one definition of stability is possible. Here we consider two of these stability criteria, socalled superstability and weak stability. We present a lineartime algorithm for finding a superstable matching if one exists, given a Stable Roommates instance with ties. This contrasts with the known NPhardness of the analogous problem under weak stability. We also extend our algorithm to cope with preference lists that are incomplete and/or partially ordered. On the other hand, for a given Stable Roommates instance with ties and incomplete lists, we show that the weakly stable matchings may be of different sizes, and the problem of finding a maximum cardinality weakly stable matching is NPhard, though approximable within a factor of 2.
Greedy Algorithms for Minimisation Problems in Random Regular Graphs
, 2001
"... . In this paper we introduce a general strategy for approximating the solution to minimisation problems in random regular graphs. We describe how the approach can be applied to the minimum vertex cover (MVC), minimum independent dominating set (MIDS) and minimum edge dominating set (MEDS) proble ..."
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Cited by 10 (4 self)
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. In this paper we introduce a general strategy for approximating the solution to minimisation problems in random regular graphs. We describe how the approach can be applied to the minimum vertex cover (MVC), minimum independent dominating set (MIDS) and minimum edge dominating set (MEDS) problems. In almost all cases we are able to improve the best known results for these problems. Results for the MVC problem translate immediately to results for the maximum independent set problem. We also derive lower bounds on the size of an optimal MIDS. 1
A 2 1/10Approximation Algorithm for a Generalization of the Weighted EdgeDominating Set Problem
 In procof "ESA '00
, 2000
"... We study the approximability of the weighted edgedominating set problem. Although even the unweighted case is NPComplete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case ..."
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Cited by 10 (6 self)
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We study the approximability of the weighted edgedominating set problem. Although even the unweighted case is NPComplete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge domination is as hard to approximate as the well studied weighted vertex cover problem, we consider a natural strategy, reducing edgedominating set to edge cover. Our main result is a simple 2 1/10approximation algorithm for the weighted edgedominating set problem, improving the existing ratio, due to a simple reduction to weighted vertex cover, of 2rWVC , where rWVC is the approximation guarantee of any polynomialtime weighted vertex cover algorithm. The best value of rWVC currently stands at 2 log log V 2 log V. Furthermore we establish that the factor of 2 1/10 is...
Edge dominating set: efficient enumerationbased exact algorithms
 Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006
, 2006
"... Abstract. We analyze edge dominating set from a parameterized perspective. More specifically, we prove that this problem is in FPT for general (weighted) graphs. The corresponding algorithms rely on enumeration techniques. In particular, we show how the use of compact representations may speed up th ..."
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Cited by 10 (1 self)
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Abstract. We analyze edge dominating set from a parameterized perspective. More specifically, we prove that this problem is in FPT for general (weighted) graphs. The corresponding algorithms rely on enumeration techniques. In particular, we show how the use of compact representations may speed up the decision algorithm. 1
Improved approximation of the stable marriage problem
 Theoretical Computer Science
, 2003
"... Abstract. While the original stable marriage problem requires all participants to rank all members of the opposite sex in a strict order, two natural variations are to allow for incomplete preference lists and ties in the preferences. Either variation is polynomially solvable, but it has recently be ..."
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Cited by 9 (5 self)
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Abstract. While the original stable marriage problem requires all participants to rank all members of the opposite sex in a strict order, two natural variations are to allow for incomplete preference lists and ties in the preferences. Either variation is polynomially solvable, but it has recently been shown to be NPhard to find a maximum cardinality stable matching when both of the variations are allowed. It is easy to see that the size of any two stable matchings differ by at most a factor of two, and so, an approximation algorithm with a factor two is trivial. In this paper, we give a first nontrivial result for the approximation with factor less than two. Our randomized algorithm achieves a factor of 10/7 for a restricted but still NPhard case, where ties occur in only men’s lists, each man writes at most one tie, and the length of ties is two. Furthermore, we show that these restrictions except for the last one can be removed without increasing the approximation ratio too much. 1
Algorithmic aspect of ktuple domination in graphs
 Taiwanese J. Math
"... Abstract. In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the ktuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies ..."
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Cited by 7 (1 self)
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Abstract. In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the ktuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the ktuple domination problem in graphs from an algorithmic point of view. In particular, we give a lineartime algorithm for the 2tuple domination problem in trees by employing a labeling method. 1.