scholarly journals Reducing the rank of a matroid

2015 ◽  
Vol Vol. 17 no.2 (Discrete Algorithms) ◽  
Author(s):  
Gwenaël Joret ◽  
Adrian Vetta
Keyword(s):  
Approximation Algorithm ◽  
Polynomial Time ◽  
Open Problem ◽  
Vertex Cover ◽  
Minimum Size ◽  
Np Hard ◽  
Rank Reduction ◽  
Reduction Problem ◽  
International Audience ◽  
Cover Problem

International audience We consider the <i>rank reduction problem</i> for matroids: Given a matroid $M$ and an integer $k$, find a minimum size subset of elements of $M$ whose removal reduces the rank of $M$ by at least $k$. When $M$ is a graphical matroid this problem is the minimum $k$-cut problem, which admits a 2-approximation algorithm. In this paper we show that the rank reduction problem for transversal matroids is essentially at least as hard to approximate as the densest $k$-subgraph problem. We also prove that, while the problem is easily solvable in polynomial time for partition matroids, it is NP-hard when considering the intersection of two partition matroids. Our proof shows, in particular, that the maximum vertex cover problem is NP-hard on bipartite graphs, which answers an open problem of B.&nbsp;Simeone.

scholarly journals Coloring and Guarding Arrangements

2013 ◽  
Vol Vol. 15 no. 3 (Combinatorics) ◽  
Author(s):  
Prosenjit Bose ◽  
Jean Cardinal ◽  
Sébastien Collette ◽  
Ferran Hurtado ◽  
Matias Korman ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Vertex Cover ◽  
Minimum Size ◽  
Vertex Cover Problem ◽  
Minimum Number ◽  
International Audience ◽  
A Cell ◽  
Cover Problem ◽  
Minimum Vertex Cover Problem ◽  
Arrangement Of Lines

Combinatorics International audience Given an arrangement of lines in the plane, what is the minimum number c of colors required to color the lines so that no cell of the arrangement is monochromatic? In this paper we give bounds on the number c both for the above question, as well as some of its variations. We redefine these problems as geometric hypergraph coloring problems. If we define $\Hlinecell$ as the hypergraph where vertices are lines and edges represent cells of the arrangement, the answer to the above question is equal to the chromatic number of this hypergraph. We prove that this chromatic number is between Ω(logn/loglogn). and O(n√). Similarly, we give bounds on the minimum size of a subset S of the intersections of the lines in A such that every cell is bounded by at least one of the vertices in S. This may be seen as a problem on guarding cells with vertices when the lines act as obstacles. The problem can also be defined as the minimum vertex cover problem in the hypergraph $\Hvertexcell$, the vertices of which are the line intersections, and the hyperedges are vertices of a cell. Analogously, we consider the problem of touching the lines with a minimum subset of the cells of the arrangement, which we identify as the minimum vertex cover problem in the $\Hcellzone$ hypergraph.

Ramsey numbers and an approximation algorithm for the vertex cover problem

1985 ◽  
Vol 22 (1) ◽  
pp. 115-123 ◽  
Author(s):  
Burkhard Monien ◽  
Ewald Speckenmeyer
Keyword(s):  
Approximation Algorithm ◽  
Vertex Cover ◽  
Ramsey Numbers ◽  
Vertex Cover Problem ◽  
Cover Problem

An approximation algorithm for the partial vertex cover problem in hypergraphs

2014 ◽  
Vol 31 (2) ◽  
pp. 846-864 ◽  
Author(s):  
Mourad El Ouali ◽  
Helena Fohlin ◽  
Anand Srivastav
Keyword(s):  
Approximation Algorithm ◽  
Vertex Cover ◽  
Vertex Cover Problem ◽  
Cover Problem

scholarly journals The resolving number of a graph Delia

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Delia Garijo ◽  
Antonio González ◽  
Alberto Márquez
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Clique Number ◽  
Metric Dimension ◽  
Np Hard ◽  
Arbitrary Graph ◽  
International Audience ◽  
Graph Parameters ◽  
Two Parameters

Graph Theory International audience We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.

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