Implementation and Comparison of Vertex Cover Problem using Various Techniques
17 Jun 2016-International Journal of Computer Applications (Foundation of Computer Science (FCS), NY, USA)-Vol. 144, Iss: 10, pp 26-31
TL;DR: Alom's Algorithm is more effective in finding MVC for undirected graphs and for weighted graphs, superior performance is attained by primal-dual based approach.
Abstract: he problem of finding Minimum Vertex Cover for graph belongs to the class of NP Complete and plays a key role in Computer Science Theory. The problems which belong to NP Complete set are not solvable in polynomial time in any known way. Since finding Minimum Vertex Cover (MVC) for a graph belongs to NP Complete class; so we are dubious to solve it in any polynomial time algorithm. Such problems are solved by algorithms which promise to give near optimum solution. In this paper we have analyzed and scrutinized such algorithms like greedy algorithm, approximation algorithm, simple genetic algorithm (GA), primal-dual based algorithm (PDB), Alom's algorithm etc. on random directed and undirected graphs and found that all the algorithms give near optimum solution with a negligible performance difference. It was also observed that out of all the above said algorithms Alom's Algorithm is more effective in finding MVC for undirected graphs and for weighted graphs, superior performance is attained by primal-dual based approach.
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TL;DR: It is shown that, with the size of the desired vertex cover as the parameter, CONNECTED VERTEX COVER and CAPACITATED VERT EX COVER are both fixed-parameter tractable while MAXIMUM PARTIAL VERTex COVER is W[1]-complete.
Abstract: Important variants of theVERTEX COVER problem (among others, CONNECTED VERTEX COVER, CAPACITATED VERTEX COVER, and MAXIMUM PARTIAL VERTEX COVER) have been intensively studied in terms of polynomial-time approximability. By way of contrast, their parameterized complexity has so far been completely open. We close this gap here by showing that, with the size of the desired vertex cover as the parameter, CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER are both fixed-parameter tractable while MAXIMUM PARTIAL VERTEX COVER is W[1]-complete. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered variants of VERTEX COVER behave very similar in terms of constant factor approximability, they display a wide range of different characteristics when investigating their parameterized complexities.
119 citations
TL;DR: The authors propose an online algorithm in which the scanned vertex is selected if and only if it has at least a nonselected already revealed neighbor with competitive ratio of ∇.
30 citations
Journal Article•
TL;DR: This paper considers the classical NP-complete VERTEX COVER problem in large graphs and proposes three algorithms that satisfy the severe constraints, and derives exact formulas giving the expected size of the solution they return.
Abstract: In this paper, we consider the classical NP-complete 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
14 citations
01 Jan 2010
TL;DR: This paper proposes three algorithms (that are variations on the same “root” list algorithms) that satisfy severe constraints of the classical NP-complete minimization Vertex Cover problem in large graphs.
Abstract: In a classical treatment of an optimization graph problem, the whole graph is available in the machine. It can be modified, updated (vertices can be marked, edges deleted, etc.) and the solution can be stored. However, many applications produce now data that are too large and are impossible to store and treat in this model. We focus in this paper on the treatment of the classical NP-complete minimization Vertex Cover problem in large graphs. The difficulty is twofold: to the intrinsic NP-completeness is added the difficulty to manipulate the input graph with severe restrictions: (1) the input graph must not be modified (integrity of the input instance), (2) the graph must be “scanned” piece by piece since it is too large to be entirely loaded in the main memory and (3) the result must be sent to an output memory once a new piece of solution is calculated. Hence, we suppose that the size and the source of the input graph impose severe treatment constraints. After modeling this situation, we show that most of the known approximation algorithms are not compatible with our model. In a second part, we propose three algorithms (that are variations on the same “root” list algorithms) that satisfy our severe constraints. In the following, we compare them by giving exact formulas expressing the expected size of the Vertex Cover constructed by the algorithms on specific designed graphs. General Terms: Algorithms, Measurement, Performance, Theory Additional
10 citations
TL;DR: This comparison paper has selected five approximation algorithms and drawn detailed experimental comparison, best available benchmarks were used for the comparison process which was to compare multiple algorithms for the same task on different aspects.
Abstract: Numerous approximation algorithms have been presented by researchers for approximation of minimum vertex cover, all of these approaches have deficiencies in one way or another. As minimum vertex cover is NP-Complete so we can’t find out optimal solution so approximation is the way left but it is very hard for someone to decide which one procedure to use, in this comparison paper we have selected five approximation algorithms and have drawn detailed experimental comparison. Best available benchmarks were used for the comparison process which was to compare multiple algorithms for the same task on different aspects. Extensive results have been provided to clarify the selection process, probability of production optimal solutions, run time complexity and approximation ratio were factors involved in the process of selection.
9 citations