Solving Connected Dominating Set Faster than 2 n

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

doi:10.1007/s00453-007-9145-z

Cited by 66 publications

(34 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The parameterized version of the problem is known to be W[2]-complete for general graphs and admits a sub-exponential time parameterized algorithm for planar graphs [8]. In general graphs the problem has also been studied in the realm of moderately exponential time algorithms leading to an algorithm with running time O(1.9407 n ) [9]. Here we provide polynomial time data reduction rules for Connected Dominating Set on planar graphs which lead to a linear kernel for the problem.…”

Section: Introductionmentioning

confidence: 99%

Linear Kernel for Planar Connected Dominating Set

Lokshtanov

Mnich

Saurabh

2009

Lecture Notes in Computer Science

View full text Add to dashboard Cite

We provide polynomial time data reduction rules for Connected Dominating Set on planar graphs and analyze these to obtain a linear kernel for the planar Connected Dominating Set problem. To obtain the desired kernel we introduce a method that we call reduce or refine. Our kernelization algorithm analyzes the input graph and either finds an appropriate reduction rule that can be applied, or zooms in on a region of the graph which is more amenable to reduction. We find this method of independent interest and believe that it will be useful to obtain linear kernels for other problems on planar graphs.

show abstract

Section: Introductionmentioning

confidence: 99%

Linear Kernel for Planar Connected Dominating Set

Lokshtanov

Mnich

Saurabh

2009

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…This problem is a natural generalization of the well studied Maximum Leaf Spanning Tree problem on connected undirected graphs [5, 7, 10-12, 14, 15, 20, 22]. Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms [14,20,22], parameterized algorithms [5,10,12], exact exponential time algorithms [11] and also combinatorial studies [7,15,16,19], the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently. Apart from [2] mentioned below, the only other paper is the very recent paper [9] that describes an O( √ opt)-approximation algorithms for DMLOB.…”

Section: Given a Digraph D A Subdigraph T Of D Is An Out-tree If T Imentioning

confidence: 99%

Better Algorithms and Bounds for Directed Maximum Leaf Problems

Alon

Fomin

Gutin

et al. 2007

FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that -every strongly connected digraph D of order n with minimum indegree at least 3 has an out-branching with at least (n/4) 1/3 − 1 leaves; -if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph is O(k log k); -it can be decided in time 2whether a strongly connected digraph on n vertices has an out-branching with at least k leaves.All improvements use properties of extremal structures obtained after applying local search and properties of some out-branching decompositions.

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Solving Connected Dominating Set Faster than 2 n

Cited by 66 publications

References 31 publications

Linear Kernel for Planar Connected Dominating Set

Linear Kernel for Planar Connected Dominating Set

Better Algorithms and Bounds for Directed Maximum Leaf Problems

Computing Optimal Steiner Trees in Polynomial Space

Contact Info

Product

Resources

About