scholarly journals Spider Covers and Their Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Filomena De Santis ◽  
Luisa Gargano ◽  
Mikael Hammar ◽  
Alberto Negro ◽  
Ugo Vaccaro
Keyword(s):  
Combinatorial Optimization ◽  
Approximation Algorithm ◽  
Greedy Algorithm ◽  
Directed Graph ◽  
Optimization Problems ◽  
Disjoint Paths ◽  
Combinatorial Optimization Problems ◽  
The Family ◽  
Minimum Number ◽  
Cover Problem

We introduce two new combinatorial optimization problems: the Maximum Spider Problem and the Spider Cover Problem; we study their approximability and illustrate their applications. In these problems we are given a directed graph , a distinguished vertex , and a family D of subsets of vertices. A spider centered at vertex s is a collection of arc-disjoint paths all starting at s but ending into pairwise distinct vertices. We say that a spider covers a subset of vertices X if at least one of the endpoints of the paths constituting the spider other than s belongs to X. In the Maximum Spider Problem the goal is to find a spider centered at s that covers the maximum number of elements of the family D. Conversely, the Spider Cover Problem consists of finding the minimum number of spiders centered at s that covers all subsets in D. We motivate the study of the Maximum Spider and Spider Cover Problems by pointing out a variety of applications. We show that a natural greedy algorithm gives a 2-approximation algorithm for the Maximum Spider Problem and a -approximation algorithm for the Spider Cover Problem.

Analyses of Simple Hybrid Algorithms for the Vertex Cover Problem

2009 ◽  
Vol 17 (1) ◽  
pp. 3-19 ◽  
Author(s):  
Tobias Friedrich ◽  
Jun He ◽  
Nils Hebbinghaus ◽  
Frank Neumann ◽  
Carsten Witt
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Hybrid Methods ◽  
Vertex Cover ◽  
Optimization Methods ◽  
Theoretical Understanding ◽  
Combinatorial Optimization Problems ◽  
Vertex Cover Problem ◽  
The Subject ◽  
Cover Problem

Hybrid methods are very popular for solving problems from combinatorial optimization. In contrast, the theoretical understanding of the interplay of different optimization methods is rare. In this paper, we make a first step into the rigorous analysis of such combinations for combinatorial optimization problems. The subject of our analyses is the vertex cover problem for which several approximation algorithms have been proposed. We point out specific instances where solutions can (or cannot) be improved by the search process of a simple evolutionary algorithm in expected polynomial time.

scholarly journals SUB-COLORING AND HYPO-COLORING INTERVAL GRAPHS

2010 ◽  
Vol 02 (03) ◽  
pp. 331-345 ◽  
Author(s):  
RAJIV GANDHI ◽  
BRADFORD GREENING ◽  
SRIRAM PEMMARAJU ◽  
RAJIV RAMAN
Keyword(s):  
Approximation Algorithm ◽  
Job Scheduling ◽  
Optimization Problems ◽  
Interval Graphs ◽  
Coloring Problem ◽  
Combinatorial Optimization Problems ◽  
Forbidden Subgraph Characterization ◽  
Color Class ◽  
Minimum Number

In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as "subroutines" for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a "forbidden subgraph" characterization of graphs with sub-chromatic number k and use this to derive a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete, and then via reduction to the max-coloring problem, show how to obtain an O( log n)-approximation algorithm for it.

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