A ($2-\varepsilon$)-approximation ratio for vertex cover problem on special graphs
محورهای موضوعی : Combinatorics, Graph theory
N. Yekezare
1
,
M. Zohrehbandian
2
,
M. Maghasedi
3
1 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
2 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
3 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
کلید واژه: Discrete optimization, Vertex cover problem, Complexity theory,
چکیده مقاله :
The vertex cover problem is a famous combinatorial problem, and its complexity has been heavily studied. It is known that it is hard to approximate to within any constant factor better than $2$, while a $2$-approximation for it can be trivially obtained. In this paper, new properties and new techniques are introduced which lead to approximation ratios smaller than $2$ on special graphs; e.g. graphs for which their maximum cut values are less than $0.999|E|$. In fact, we produce a ($1.9997$)-approximation ratio on graph $G$, where the ($0.878$)-approximation algorithm of the Goemans-Williamson for the maximum cut problem on $G$ produces a value smaller than $0.877122|E|$.
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