Approximation algorithms for the maximally balanced connected graph tripartition problem

2020 ◽  
Author(s):  
Guangting Chen ◽  
Yong Chen ◽  
Zhi-Zhong Chen ◽  
Guohui Lin ◽  
Tian Liu ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Connected Graph

TWO ALGORITHMS FOR CONNECTED r-HOP k-DOMINATING SET

2009 ◽  
Vol 01 (04) ◽  
pp. 485-498 ◽  
Author(s):  
ZHAO ZHANG ◽  
QINGHAI LIU ◽  
DEYING LI
Keyword(s):  
Wireless Sensor Network ◽  
Approximation Algorithms ◽  
Sensor Network ◽  
Connected Graph ◽  
Dominating Set ◽  
Connected Dominating Set ◽  
Performance Ratio ◽  
Wireless Sensor ◽  
Vertex Set

A vertex set D of a connected graph G is a (k, r)-connected dominating set ((k, r)-CDS) if every vertex in V(G)\D is at most r-hops away from at least k vertices in D. Finding a minimum (k, r)-CDS has wireless sensor network as its background. In this paper, we give two approximation algorithms to compute a minimum (k, r)-CDS, which improves previous works in regard of performance ratio.

scholarly journals Approximation Algorithms for Maximally Balanced Connected Graph Partition

Algorithmica ◽  
2021 ◽  
Author(s):  
Yong Chen ◽  
Zhi-Zhong Chen ◽  
Guohui Lin ◽  
Yao Xu ◽  
An Zhang
Keyword(s):  
Approximation Algorithms ◽  
Connected Graph ◽  
Graph Partition

Linear-Time Approximation Algorithms for the Max Cut Problem

1993 ◽  
Vol 2 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Nguyen van Ngoc ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Approximation Algorithms ◽  
Lower Bounds ◽  
Connected Graph ◽  
Linear Time ◽  
Worst Case ◽  
Time Approximation ◽  
Bipartite Subgraph ◽  
Max Cut Problem ◽  
Multiple Edges

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

scholarly journals On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

2021 ◽  
Author(s):  
Waldo Gálvez ◽  
Fabrizio Grandoni ◽  
Afrouz Jabal Ameli ◽  
Krzysztof Sornat
Keyword(s):  
Approximation Algorithms ◽  
Connected Graph ◽  
Minimum Size ◽  
Single Cycle ◽  
Integrality Gap ◽  
Connectivity Augmentation ◽  
Special Case ◽  
Better Than

AbstractIn the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial $\left (\frac {3}{2}+\varepsilon \right )$ 3 2 + ε -approximation for CycAP, for any constant ε > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

scholarly journals Approximation and inapproximability results on balanced connected partitions of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Frédéric Chataigner ◽  
Liliane R. B. Salgado ◽  
Yoshiko Wakabayashi
Keyword(s):  
Approximation Algorithms ◽  
Positive Integer ◽  
Approximation Algorithm ◽  
Weight Function ◽  
Connected Graph ◽  
Strong Sense ◽  
Connected Graphs ◽  
International Audience ◽  
Arbitrary Graphs ◽  
Balanced Connected Partition

Graphs and Algorithms International audience Let G=(V,E) be a connected graph with a weight function w: V \to \mathbbZ₊, and let q ≥q 2 be a positive integer. For X⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a q-partition P=(V₁,V₂, \ldots, V_q) of V such that G[V_i] is connected (1≤q i≤q q) and P maximizes \rm min\w(V_i): 1≤q i≤q q\. This problem is called \textitMax Balanced Connected q-Partition and is denoted by BCP_q. We show that for q≥q 2 the problem BCP_q is NP-hard in the strong sense, even on q-connected graphs, and therefore does not admit a FPTAS, unless \rm P=\rm NP. We also show another inapproximability result for BCP₂ on arbitrary graphs. On q-connected graphs, for q=2 the best result is a \frac43-approximation algorithm obtained by Chleb\'ıková; for q=3 and q=4 we present 2-approximation algorithms. When q is not fixed (it is part of the instance), the corresponding problem is called \textitMax Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless \rm P=\rm NP.

scholarly journals Approximation Algorithms for Connected Graph Factors of Minimum Weight

2016 ◽  
Vol 62 (2) ◽  
pp. 441-464 ◽  
Author(s):  
Kamiel Cornelissen ◽  
Ruben Hoeksma ◽  
Bodo Manthey ◽  
N. S. Narayanaswamy ◽  
C. S. Rahul ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Connected Graph ◽  
Minimum Weight ◽  
Graph Factors

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Sign in / Sign up

Export Citation Format

Share Document