A Parameterized Measure-and-ConquerAnalysis for Finding a k-Leaf Spanning Treein an Undirected Graph

Daniel Binkele-Raible; Henning Fernau

doi:10.46298/dmtcs.1256

A Parameterized Measure-and-ConquerAnalysis for Finding a k-Leaf Spanning Treein an Undirected Graph

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.1256 ◽

2014 ◽

Vol Vol. 16 no. 1 (Discrete Algorithms) ◽

Author(s):

Daniel Binkele-Raible ◽

Henning Fernau

Keyword(s):

Spanning Tree ◽

Undirected Graph ◽

Standard Measure ◽

Algorithm Analysis ◽

Np Hard ◽

Running Time ◽

Discrete Algorithms ◽

Parameterized Algorithmics ◽

International Audience ◽

Number Of Leaves

Discrete Algorithms International audience The problem of finding a spanning tree in an undirected graph with a maximum number of leaves is known to be NP-hard. We present an algorithm which finds a spanning tree with at least k leaves in time O*(3.4575k) which improves the currently best algorithm. The estimation of the running time is done by using a non-standard measure. The present paper is one of the still few examples that employ the Measure & Conquer paradigm of algorithm analysis in the area of Parameterized Algorithmics.

The Problem of Scheduling Parallel Computations of Multibody Dynamic Analysis Is NP-Hard

Volume 2A: 24th Biennial Mechanisms Conference ◽

10.1115/96-detc/mech-1150 ◽

1996 ◽

Author(s):

Jin-Fan Liu ◽

Karim A. Abdel-Malek

Keyword(s):

Dynamic Analysis ◽

Spanning Tree ◽

Minimum Weight ◽

Computational Cost ◽

Parallel Computations ◽

Np Hard ◽

Multibody Dynamic ◽

Np Hard Problem ◽

Np Complete ◽

Minimum Weight Spanning Tree

Abstract A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. The problems of shortest critical branch minimum weight spanning tree (SCBMWST) and the minimum weight shortest critical branch spanning tree (MWSCBST) are also presented. Both problems are shown to be NP-hard by proving that the bounded critical branch bounded weight spanning tree (BCBBWST) problem is NP-complete. It is also shown that the minimum computational cost spanning tree (MCCST) is at least as hard as SCBMWST or MWSCBST problems, hence itself an NP-hard problem. A heuristic approach to solving these problems is developed and implemented, and simulation results are discussed.

On the number of leaves of a euclidean minimal spanning tree

Journal of Applied Probability ◽

10.2307/3214207 ◽

1987 ◽

Vol 24 (4) ◽

pp. 809-826 ◽

Cited By ~ 18

Author(s):

J. Michael Steele ◽

Lawrence A. Shepp ◽

William F. Eddy

Keyword(s):

Spanning Tree ◽

Spanning Trees ◽

Random Variables ◽

Minimal Spanning Tree ◽

Absolutely Continuous ◽

Mean And Variance ◽

Number Of Leaves ◽

The Mean ◽

Vertex Degrees ◽

Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

Consensus Problem of Singular Multi-Agent Systems with Continuous-Time and Fixed Topology

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.278-280.1687 ◽

2013 ◽

Vol 278-280 ◽

pp. 1687-1691

Author(s):

Tong Qiang Jiang ◽

Jia Wei He ◽

Yan Ping Gao

Keyword(s):

Continuous Time ◽

Spanning Tree ◽

Undirected Graph ◽

Necessary Condition ◽

Multi Agent Systems ◽

Agent Systems ◽

Multi Agent ◽

Fixed Topology ◽

The Stability ◽

Sufficient And Necessary Condition

The consensus problems of two situations for singular multi-agent systems with fixed topology are discussed: directed graph without spanning tree and the disconnected undirected graph. A sufficient and necessary condition is obtained by applying the stability theory and the system is reachable asymptotically. But for normal systems, this can’t occur in upper two situations. Finally a simulation example is provided to verify the effectiveness of our theoretical result.

Parameterized Algorithmics for Finding Exact Solutions of NP-Hard Biological Problems

Methods in Molecular Biology - Bioinformatics ◽

10.1007/978-1-4939-6613-4_20 ◽

2016 ◽

pp. 363-402 ◽

Cited By ~ 1

Author(s):

Falk Hüffner ◽

Christian Komusiewicz ◽

Rolf Niedermeier ◽

Sebastian Wernicke

Keyword(s):

Exact Solutions ◽

Np Hard ◽

Parameterized Algorithmics

The Minimum Stretch Spanning Tree Problem for Hamming Graphs and Higher-Dimensional Grids

Journal of Interconnection Networks ◽

10.1142/s0219265920500048 ◽

2020 ◽

Vol 20 (01) ◽

pp. 2050004

Author(s):

LAN LIN ◽

YIXUN LIN

Keyword(s):

Communication Networks ◽

Spanning Tree ◽

Optimization Problem ◽

Maximum Distance ◽

Np Hard ◽

Graph Invariant ◽

Minimum Value ◽

Higher Dimensional ◽

Hamming Graphs

The minimum stretch spanning tree problem for a graph G is to find a spanning tree T of G such that the maximum distance in T between two adjacent vertices is minimized. The minimum value of this optimization problem gives rise to a graph invariant σ(G), called the tree-stretch of G. The problem has been proved NP-hard. In this paper we present a general approach to determine the exact values σ(G) for a series of typical graphs arising from communication networks, such as Hamming graphs and higher-dimensional grids (including hypercubes).

On the number of leaves of a euclidean minimal spanning tree

Journal of Applied Probability ◽

10.1017/s0021900200116705 ◽

1987 ◽

Vol 24 (04) ◽

pp. 809-826 ◽

Cited By ~ 2

Author(s):

J. Michael Steele ◽

Lawrence A. Shepp ◽

William F. Eddy

Keyword(s):

Spanning Tree ◽

Spanning Trees ◽

Random Variables ◽

Minimal Spanning Tree ◽

Absolutely Continuous ◽

Mean And Variance ◽

Number Of Leaves ◽

The Mean ◽

Vertex Degrees ◽

Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

THE FIRST APPROXIMATED DISTRIBUTED ALGORITHM FOR THE MINIMUM DEGREE SPANNING TREE PROBLEM ON GENERAL GRAPHS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054104002571 ◽

2004 ◽

Vol 15 (03) ◽

pp. 507-516 ◽

Cited By ~ 5

Author(s):

LÉLIA BLIN ◽

FRANCK BUTELLE

Keyword(s):

Distributed Algorithm ◽

Spanning Tree ◽

Time Complexity ◽

Minimum Degree ◽

Np Hard ◽

Message Complexity ◽

General Graphs

In this paper we present the first distributed algorithm on general graphs for the Minimum Degree Spanning Tree problem. The problem is NP-hard in sequential. Our algorithm give a Spanning Tree of a degree at most 1 from the optimal. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves O(|V|) time complexity and O(|V||E|) message complexity.

An FPT Algorithm for the Correlation Clustering Problem

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.474-476.924 ◽

2011 ◽

Vol 474-476 ◽

pp. 924-927 ◽

Cited By ~ 1

Author(s):

Xiao Xin

Keyword(s):

Approximation Algorithms ◽

Polynomial Time ◽

Undirected Graph ◽

Fixed Parameter Tractable ◽

Correlation Clustering ◽

Time Approximation ◽

Running Time ◽

Clustering Problem ◽

Fpt Algorithm ◽

Fixed Parameter

Given an undirected graph G=(V, E) with real nonnegative weights and + or – labels on its edges, the correlation clustering problem is to partition the vertices of G into clusters to minimize the total weight of cut + edges and uncut – edges. This problem is APX-hard and has been intensively studied mainly from the viewpoint of polynomial time approximation algorithms. By way of contrast, a fixed-parameter tractable algorithm is presented that takes treewidth as the parameter, with a running time that is linear in the number of vertices of G.

Computing a minimum-dilation spanning tree is NP-hard

Computational Geometry ◽

10.1016/j.comgeo.2007.12.001 ◽

2008 ◽

Vol 41 (3) ◽

pp. 188-205 ◽

Cited By ~ 12

Author(s):

Otfried Cheong ◽

Herman Haverkort ◽

Mira Lee

Keyword(s):

Spanning Tree ◽

Np Hard ◽

Minimum Dilation

Leafy spanning k-forests

10.5753/etc.2021.16375 ◽

2021 ◽

Author(s):

Cristina G. Fernandes ◽

Carla N. Lintzmayer ◽

Mário César San Felice

Keyword(s):

Positive Integer ◽

Approximation Algorithm ◽

Best Approximation ◽

Np Hard ◽

Connected Graphs ◽

Spanning Forest ◽

The Best Approximation ◽

Number Of Leaves

We denote by Maximum Leaf Spanning k-Forest the problem of, given a positive integer k and a graph G with at most k components, finding a spanning forest in G with at most k components and the maximum number of leaves. A leaf in a forest is defined as a vertex of degree at most one. The case k = 1 for connected graphs is known to be NP-hard, and is well studied in the literature, with the best approximation algorithm proposed more than 20 years ago by Solis-Oba. The best known approximation algorithm for Maximum Leaf Spanning k-Forest with a slightly different leaf definition is a 3-approximation based on an approach by Lu and Ravi for the k = 1 case. We extend the algorithm of Solis-Oba to achieve a 2-approximation for Maximum Leaf Spanning k-Forest.