Measure and Conquer: Domination – A Case Study

Fomin, Fedor V.; Grandoni, Fabrizio; Kratsch, Dieter

doi:10.1007/11523468_16

Cited by 102 publications

(107 citation statements)

References 20 publications

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“…We refer to [9,10] for a reference on Measure & Conquer. Even though Measure & Conquer has been applied to several problems to obtain exact exponential time algorithms, its applicability in obtaining parameterized algorithm has been limited to an algorithm for 3-Hitting Set by Wahlström [20].…”

Section: Theoremmentioning

confidence: 99%

Even Faster Algorithm for Set Splitting!

Lokshtanov

Saurabh

2009

Parameterized and Exact Computation

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In the p-Set Splitting problem we are given a universe U , a family F of subsets of U and a positive integer k and the objective is to find a partition of U into W and B such that there are at least k sets in F that have non-empty intersection with both B and W . In this paper we study p-Set Splitting from the view point of kernelization and parameterized algorithms. Given an instance (U, F, k) of p-Set Splitting, our kernelization algorithm obtains an equivalent instance with at most 2k sets and k elements in polynomial time. Finally, we give a fixed parameter tractable algorithm for p-Set Splitting running in time O(1.9630 k + N ), where N is the size of the instance. Both our kernel and our algorithm improve over the best previously known results. Our kernelization algorithm utilizes a classical duality theorem for a connectivity notion in hypergraphs. We believe that the duality theorem we make use of could become an important tool in obtaining kernelization algorithms.

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Section: Theoremmentioning

confidence: 99%

Even Faster Algorithm for Set Splitting!

Lokshtanov

Saurabh

2009

Parameterized and Exact Computation

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“…There exists at least one vertex j ∈ A such thatd A (j) 1, or else a pure set covering problem where the set F corresponds to the universe U of elements and the set A corresponds to the collection S of the (nonempty) subsets of U and the aim is to determine a minimum cardinality subcollection S ′ ⊆ S which covers U . This set covering problem is known to be solvable to optimality in O * (1.2301 |A|+|F | ) time ( [2]). But, as |A| + |F | n this is not superior to O * (1.2301 n ) time.…”

Section: Max Dominating Cliquementioning

confidence: 99%

“…Recently, [7] have proposed a branching algorithm that, according to a measure and conquer analysis [2], solves min dominating clique with polynomial space and running time O * (1.3387 |V | ), and another one that requires O * (1.3234 |V | ) time and space. Naturally, these algorithms also solve existing dominating clique.…”

Section: Introductionmentioning

confidence: 99%

Exact Algorithms for Dominating Clique Problems

Bourgeois

Croce

Escoffier

et al. 2009

Algorithms and Computation

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We handle in this paper three dominating clique problems, namely, the decision problem itself when one asks if there exists a dominating clique in a graph G and two optimization versions where one asks for a maximum-and a minimum-size dominating clique, if any. For the three problems we propose optimal algorithms with provably worst-case upper bounds improving existing ones by (D. Kratsch and M. Liedloff, An exact algorithm for the minimum dominating clique problem, Theoretical Computer Science 385(1-3), pp. [226][227][228][229][230][231][232][233][234][235][236][237][238][239][240] 2007). We then settle all the three problems in sparse and dense graphs also providing improved upper running time bounds.

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“…For this reason we use the Measure & Conquer analytical technique described in [21] (see also [17,18]), which is based on the quasiconvex analysis of multivariate recurrences by Eppstein [13]. The basic idea is designing a convenient (non-trivial) measure of the size of the problem.…”

Section: Introductionmentioning

confidence: 99%

“…The running time obtained with respect to the refined measure is eventually turned into the equivalent running time in terms of some standard measure (typically the number of nodes or edges for graph problems). Measure & Conquer has been successfully applied to the design of exact algorithms for coloring [12], independent set [18], dominating set [17,22,27,28], cubic-TSP [14], feedback vertex set [16], and maximum leaf spanning tree [20], among others. As it will be clearer from the analysis, a convenient measure in our case is a linear combination of the number n of nodes and number n − k of non-terminals in the graph.…”

Section: Introductionmentioning

confidence: 99%

Computing Optimal Steiner Trees in Polynomial Space

Fomin

Grandoni

Kratsch³

et al. 2012

Algorithmica

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Given an n-node edge-weighted graph and a subset of k terminal nodes, the NP-hard (weighted) Steiner tree problem is to compute a minimum-weight tree which spans the terminals. All the known algorithms for this problem which improve on trivial O(1.62 n )-time enumeration are based on dynamic programming, and require exponential space.Motivated by the fact that exponential-space algorithms are typically impractical, in this paper we address the problem of designing faster polynomial-space algorithms. Our first contribution is a simple O((27/4) k n O(log k) )-time polynomial-space algorithm for the problem. This algorithm is based on a variant of the classical tree-separator theorem: every Steiner tree has a node whose removal partitions the tree in two forests, containing at most 2k/3 terminals each. Exploiting separators of logarithmic size which evenly partition the terminals, we are able to reduce the running time to O(4 k n O(log 2 k) ). This improves on trivial enumeration for roughly k < n/3, which covers most of the cases of practical interest. Combining the latter algorithm (for small k) with trivial enumeration (for large k) we obtain a O(1.59 n )-time polynomial-space algorithm for the weighted Steiner tree problem.As a second contribution of this paper, we present a O(1.55 n )-time polynomial-space algorithm for the cardinality version of the problem, where all edge weights are one. This result is based on a improved branching strategy. The refined branching is based on a charging mechanism which shows that, for large values of k, convenient local configurations of terminals and non-terminals exist. The analysis of the algorithm relies on the Measure & Conquer approach: the non-standard measure used here is a linear combination of the number of nodes and number of non-terminals. Using a recent result in [Nederlof'09], the running time can be reduced to O(1.36 n ). The previous best algorithm for the cardinality case runs in O(1.42 n ) time and exponential space.

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Measure and Conquer: Domination – A Case Study

Cited by 102 publications

References 20 publications

Even Faster Algorithm for Set Splitting!

Even Faster Algorithm for Set Splitting!

Exact Algorithms for Dominating Clique Problems

Computing Optimal Steiner Trees in Polynomial Space

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