Approximation Algorithms for Maximally Balanced Connected Graph Partition

We consider the following clustering problems: given an undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

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TWO ALGORITHMS FOR CONNECTED r-HOP k-DOMINATING SET

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830909000361 ◽

2009 ◽

Vol 01 (04) ◽

pp. 485-498 ◽

Cited By ~ 16

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.

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Balanced Connected Graph Partition

Algorithms and Discrete Applied Mathematics - Lecture Notes in Computer Science ◽

10.1007/978-3-030-67899-9_38 ◽

2021 ◽

pp. 487-499

Author(s):

Satyabrata Jana ◽

Supantha Pandit ◽

Sasanka Roy

Keyword(s):

Connected Graph ◽

Graph Partition

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Linear-Time Approximation Algorithms for the Max Cut Problem

Combinatorics Probability Computing ◽

10.1017/s0963548300000596 ◽

1993 ◽

Vol 2 (2) ◽

pp. 201-210 ◽

Cited By ~ 11

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.

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On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

Theory of Computing Systems ◽

10.1007/s00224-020-10025-6 ◽

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.

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Approximation and inapproximability results on balanced connected partitions of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.384 ◽

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.

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