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12
Approximation Algorithms for Connected Dominating Sets
 Algorithmica
, 1996
"... The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, whe ..."
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Cited by 376 (9 self)
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H (\Delta)) are presented, where \Delta is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the ...
On Approximation Algorithms for the Minimum Satisfiability Problem
 Information Processing Letters
, 1996
"... this paper, our focus is on deterministic approximation algorithms for the MINSAT problem. From now on, we will use the word `heuristic' to mean a deterministic approximation algorithm which runs in polynomial time. Note that when the clauses are of size \Theta(n), where n is the number of vari ..."
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Cited by 25 (0 self)
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this paper, our focus is on deterministic approximation algorithms for the MINSAT problem. From now on, we will use the word `heuristic' to mean a deterministic approximation algorithm which runs in polynomial time. Note that when the clauses are of size \Theta(n), where n is the number of variables, the heuristic analyzed in [18] provides only a weak performance guarantee of \Theta(n). We present a simple approximationpreserving reduction from MINSAT to the minimum vertex cover (MINVC) problem. This reduction, in conjunction with known heuristics for the MINVC problem (see for example, [8,22]), yields a heuristic with a performance guarantee of 2 for MINSAT, thus improving the result of Kohli et al. [18]. We also show that MINSAT is as hard to approximate as MINVC; that is, if there is a heuristic with a performance guarantee ae for MINSAT, then there is a heuristic with the same performance guarantee ae for MINVC. Moreover, we show that this result holds even for MINSAT instances defined by Horn formulas. It has been conjectured in [12] that no polynomial approximation algorithm can provide a performance guarantee of 2 \Gamma ffl for any fixed ffl ? 0 for MINVC unless P = NP. Thus, our result provides an indication of the difficulty involved in devising a heuristic with a performance guarantee better than 2 for MINSAT.
Parameterized Complexity of Vertex Cover Variants
, 2006
"... Important variants of the Vertex Cover problem (among others, Connected Vertex Cover, Capacitated Vertex Cover, and Maximum ..."
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Cited by 23 (5 self)
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Important variants of the Vertex Cover problem (among others, Connected Vertex Cover, Capacitated Vertex Cover, and Maximum
Complexity and approximation results for the connected vertex cover problem
"... We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of graphs wher ..."
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Cited by 11 (0 self)
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We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).
Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms
, 2006
"... We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of ..."
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Cited by 9 (2 self)
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We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has least t vertices (edges). These denitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NPcompleteness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with nding the minimum size of a ttotal vertex cover, ttotal edge cover and connected vertex cover, in particular improving on a previous FPT algorithm for the latter problem.
CONNECTED VERTEX COVERS IN DENSE GRAPHS
"... Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parame ..."
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Cited by 5 (1 self)
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Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worstcase ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2/(1 + ε) in graphs with average degree εn, and give a family of neartight examples. Key words: approximation algorithm, vertex cover, connected vertex cover, dense graph. 1.
On the optimal placement of wavelength converters in WDM networks
"... An important goal of the design of wavelength division multiplexing (WDM) networks is to use less wavelengths to serve more communication needs. According to the wavelength conflict rule, we know that the number of wavelengths required in a WDM network is at least equal to the maximal number of chan ..."
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Cited by 3 (0 self)
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An important goal of the design of wavelength division multiplexing (WDM) networks is to use less wavelengths to serve more communication needs. According to the wavelength conflict rule, we know that the number of wavelengths required in a WDM network is at least equal to the maximal number of channels over a fiber (called maximal link load) in the network. By placing wavelength converters at some nodes in the network, the number of wavelengths needed can be made equal to the maximal link load. In this paper, we study the problem of placing the minimal number of converters in a network to achieve that the number of wavelengths in use is equal to the maximal link load. For duplex communication channels, we prove that the optimal solution can be obtained in polynomialtime. For unidirectional communication channels, which was proved to be NPcomplete, we develop a set of lemmas which lead to a simple approximation algorithm
Analysis and Comparison of Three Algorithms for the Vertex Cover Problem on Large Graphs with Low Memory Capacities. 1
"... In this paper, we consider the classical NPcomplete VERTEX COVER problem in large graphs. We assume that the size and the access to the input graph impose the following constraints: (1) the input graph must not be modified (integrity of the input instance), (2) the computer running the algorithm ha ..."
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Cited by 2 (1 self)
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In this paper, we consider the classical NPcomplete VERTEX COVER problem in large graphs. We assume that the size and the access to the input graph impose the following constraints: (1) the input graph must not be modified (integrity of the input instance), (2) the computer running the algorithm has a memory of limited size (compared to the graph) and (3) the result must be sent to an output memory once a new piece of solution is calculated. Despite the severe constraints of the model, we propose three algorithms that satisfy them. We derive exact formulas giving the expected size of the solution they return. This allows us to compare them, in an analytic way. Then, we consider their complexities. We give exact formulas expressing the expected number of requests they perform on the input graph. Again, we compare them analytically. For both comparisons, we show that none of them is better than the two others. The formulas we give can help users to estimate the best balance between quality of the solution and performance. Key words: large graphs, vertex cover, mean analysis of algorithms 1.
Approximation Algorithms for Connected Dominating Sets
, 1998
"... The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minim ..."
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(�) + 2 and H(�) + 2 are presented, where � is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn + 1) ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c + 1)H(�) + c − 1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644).
Placement of Wavelength Converters for Minimal Wavelength Usage in WDM Networks
"... Abstract—An important goal of the design of WDM (wavelength division multiplexing) networks is to use less wavelengths to serve more communication needs. According to the wavelength conflict rule, we know that the number of wavelengths required in a WDM network is at least equal to the maximal numb ..."
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Abstract—An important goal of the design of WDM (wavelength division multiplexing) networks is to use less wavelengths to serve more communication needs. According to the wavelength conflict rule, we know that the number of wavelengths required in a WDM network is at least equal to the maximal number of channels over a fiber (called maximal link load) in the network. By placing wavelength converters at some nodes in the network, the number of wavelengths needed can be made equal to the maximal link load. In this paper we study the problem of placing the minimal number of converters in a network to achieve that the number of wavelengths in use is equal to the maximal link load. For duplex communication channels, we prove that an optimal solution can be obtained in polynomialtime. For unidirectional communication channels, which was proved to be NPcomplete, we develop a set of lemmas which lead to an efficient approximation algorithm whose approximation ratio is two. I.