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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.
Network Design via Core Detouring for Problems Without a Core
"... Some of the currently bestknown approximation algorithms for network design are based on random sampling. One of the key steps of such algorithms is connecting a set of source nodes to a random subset of them. In a recent work [Eisenbrand, Grandoni, Rothvoß, Schäfer  SODA’08], a new technique, c ..."
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Cited by 7 (2 self)
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Some of the currently bestknown approximation algorithms for network design are based on random sampling. One of the key steps of such algorithms is connecting a set of source nodes to a random subset of them. In a recent work [Eisenbrand, Grandoni, Rothvoß, Schäfer  SODA’08], a new technique
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
Measure and Conquer: A Simple O(2^0.288n) Independent Set Algorithm
"... For more than 30 years DavisPutnamstyle exponentialtime backtracking algorithms have been the most common tools used for finding exact solutions of NPhard problems. Despite of that, the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds. The ..."
Abstract

Cited by 45 (5 self)
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For more than 30 years DavisPutnamstyle exponentialtime backtracking algorithms have been the most common tools used for finding exact solutions of NPhard problems. Despite of that, the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds. The “Measure and Conquer” approach is one of the recent attempts to step beyond such limitations. The approach is based on the choice of the measure of the subproblems recursively generated by the algorithm considered; this measure is used to lower bound the progress made by the algorithm at each branching step. A good choice of the measure can lead to a significantly better worst case time analysis. In this paper we apply “Measure and Conquer ” to the analysis of a very simple backtracking algorithm solving the wellstudied maximum independent set problem. The result of the analysis is striking: the running time of the algorithm is O(2 0.288n), which is competitive with the current best time bounds obtained with far more complicated algorithms (and naive analysis). Our example shows that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.
Thresholdbased mechanisms to discriminate transient from intermittent faults
 IEEE TRANSACTIONS ON COMPUTERS
"... AbstractÐThis paper presents a class of countandthreshold mechanisms, collectively named count, which are able to discriminate between transient faults and intermittent faults in computing systems. For many years, commercial systems have been using transient fault discrimination via thresholdbas ..."
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Cited by 44 (9 self)
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AbstractÐThis paper presents a class of countandthreshold mechanisms, collectively named count, which are able to discriminate between transient faults and intermittent faults in computing systems. For many years, commercial systems have been using transient fault discrimination via thresholdbased techniques. We aim to contribute to the utility of countandthreshold schemes, by exploring their effects on the system. We adopt a mathematically defined structure, which is simple enough to analyze by standard tools. count is equipped with internal parameters that can be tuned to suit environmental variables (such as transient fault rate, intermittent fault occurrence patterns). We carried out an extensive behavior analysis for two versions of the countandthreshold scheme, assuming, first, exponentially distributed fault occurrencies and, then, more realistic fault patterns. Index TermsÐFault discrimination, thresholdbased identification, transient and intermittent faults, modeling and evaluation, fault diagnosis. æ 1
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.
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.
Results 1  10
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