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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 59 (21 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. ..."
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Cited by 51 (11 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 46 (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 28 (2 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 27 (9 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.
Combinatorial bounds via measure and conquer: Boundings minimal dominating sets and applications
 PRELIM.VERSION IN PROC. 16TH ISAAC
, 2006
"... We provide an algorithm listing all minimal dominating sets of a graph on n vertices in time O(1.7159n). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on n vertices is at most 1.7159n, thus improving on the trivial O(2n / √ n) boun ..."
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Cited by 20 (4 self)
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We provide an algorithm listing all minimal dominating sets of a graph on n vertices in time O(1.7159n). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on n vertices is at most 1.7159n, thus improving on the trivial O(2n / √ n) bound. Our result makes use of the measure and conquer technique which was recently developed in the area of exact algorithms. Based on this result, we derive an O(2.8718n) algorithm for the domatic number problem.
On Two Techniques of Combining Branching and Treewidth
 ALGORITHMICA (2009) 54: 181–207
, 2009
"... 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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Cited by 19 (3 self)
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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
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 17 (6 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.
On Problems as Hard as CNFSat
, 2012
"... Exact exponential time algorithms for NPhard problems have thrived over the last decade. Nontrivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3–CNFSat, that is, satisfiability of 3CNF formulas. For some ba ..."
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Cited by 17 (4 self)
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Exact exponential time algorithms for NPhard problems have thrived over the last decade. Nontrivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3–CNFSat, that is, satisfiability of 3CNF formulas. For some basic problems, however, there has been no progress over their trivial solution. For others nontrivial solutions have been found, but improving these algorithms further seems to be out of reach. The CNFSat problem is the canonical example of a problem for which the brute force 2 n n O(1) time algorithm remains the best known. The assumption that k–CNFSat requires 2 n time in the worst case when k grows to infinity is known as the strong exponential time hypothesis (SETH) of Impagliazzo and Paturi. In this paper we reveal connections between wellstudied problems, and show that improving over the currently best known algorithms for several of them would violate SETH. Specifically, we show that for every ɛ < 1, an O(2 ɛn) time algorithm for Hitting Set, Set Splitting or NAESat would violate SETH. Here n is the number of elements (or