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Measure and conquer: domination  a case study
 PROCEEDINGS OF THE 32ND INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2005), SPRINGER LNCS
, 2005
"... DavisPutnamstyle exponentialtime backtracking algorithms are the most common algorithms used for finding exact solutions of NPhard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure ..."
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Cited by 47 (20 self)
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DavisPutnamstyle exponentialtime backtracking algorithms are the most common algorithms used for finding exact solutions of NPhard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure is used to lower bound the progress made by the algorithm at each branching step. For the last 30 years the research on exact algorithms has been mainly focused on the design of more and more sophisticated algorithms. However, measures used in the analysis of backtracking algorithms are usually very simple. In this paper we stress that a more careful choice of the measure can lead to significantly better worst case time analysis. As an example, we consider the minimum dominating set problem. The currently fastest algorithm for this problem has running time O(2 0.850n) on nnodes graphs. By measuring the progress of the (same) algorithm in a different way, we refine the time bound to O(2 0.598n). A good choice of the measure can provide such a (surprisingly big) improvement; this suggests that the running time of many other exponentialtime recursive algorithms is largely overestimated because of a “bad” choice of the measure.
A measure & conquer approach for the analysis of exact algorithms
, 2007
"... For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase running time bounds. Mot ..."
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Cited by 29 (8 self)
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For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase running time bounds. Motivated by this we use an approach, that we call “Measure & Conquer”, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worstcase time analysis. In order to show the potentialities of Measure & Conquer, we consider two wellstudied NPhard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis). Our examples
Incompressibility through Colors and IDs
"... In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown t ..."
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Cited by 24 (5 self)
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In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [15]. With few exceptions, all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All our results are under the assumption that the polynomial hierarchy does not collapse to the third level. • We show that the Steiner Tree problem parameterized by the number of terminals and solution size, and the Connected Vertex Cover and Capacitated Vertex Cover problems do not admit a polynomial kernel. The two latter results are surprising because the closely related Vertex Cover problem admits a kernel of size 2k.
FPT is Ptime extremal structure I
 Algorithms and Complexity in Durham 2005, Proceedings of the first ACiD Workshop, volume 4 of Texts in Algorithmics
, 2005
"... We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem. ..."
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Cited by 18 (1 self)
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We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem.
Solving Connected Dominating Set Faster than 2^n
, 2006
"... In the connected dominating set problem we are given an nnode undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of le ..."
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Cited by 18 (8 self)
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In the connected dominating set problem we are given an nnode undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2 n) algorithm that enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such a difference is not surprising, since connectivity is a global property, and nonlocal problems are typically much harder to solve exactly. In this paper we break the 2 n barrier, by presenting a simple O(1.9407 n) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique.
Bounding the number of minimal dominating sets: a measure and conquer approach
 IN PROCEEDINGS OF THE 16TH ANNUAL INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION (ISAAC 2005
, 2005
"... We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697 n, thus improving on the trivial O(2 n / √ n) bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an O(1.7697 n) listing algorithm. Based ..."
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Cited by 14 (5 self)
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We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697 n, thus improving on the trivial O(2 n / √ n) bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an O(1.7697 n) listing algorithm. Based on this result, we derive an O(2.8805 n) algorithm for the domatic number problem, and an O(1.5780 n) algorithm for the minimumweight dominating set problem. Both algorithms improve over the previous algorithms.
Algorithms Based on the Treewidth Of Sparse Graphs
 IN PROCEEDINGS OF THE 31ST INTERNATIONAL WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE (WG 2005), LNCS
, 2005
"... We prove that given a graph, one can efficiently find a set of no more than m/5.217 + 1 nodes whose removal yields a partial twotree. As an ..."
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Cited by 14 (0 self)
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We prove that given a graph, one can efficiently find a set of no more than m/5.217 + 1 nodes whose removal yields a partial twotree. As an
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
An exact 2.9416 n algorithm for the three domatic number problem
 In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science
, 2005
"... The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NPcomplete, no polynomialtime algorithm is known for it. The naive determini ..."
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Cited by 8 (2 self)
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The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NPcomplete, no polynomialtime algorithm is known for it. The naive deterministic algorithm for this problem runs in time 3 n, up to polynomial factors. In this paper, we design an exact deterministic algorithm for this problem running in time 2.9416 n. Thus, our algorithm can handle problem instances of larger size than the naive algorithm in the same amount of time. We also present another deterministic and a randomized algorithm for this problem that both have an even better performance for graphs with small maximum degree. Key words: 1