### On approximating a vertex cover for planar graphs

R. Bar-Yehuda, S. Even
1982 Proceedings of the fourteenth annual ACM symposium on Theory of computing - STOC '82
The approximation problem for vertex cover of n-vertex planar graphs is treated. Two results are presented: (i) A linear time approximation algorithm for which the (error) performance bound is 2/3. (2) An 0(n log n) time approximation scheme. i. INTRODUCTION Let G(V,E) be a simple undirected graph. A subset of vertices is called a vertex cover if every edge has at least one endpoint in the subset. The vertex cover problem is, given G, find a vertex cover of minimum cardinality. The vertex cover
more » ... problem is known to be NP-hard, even if the graphs are restricted to be cubic or planar with maximum vertex degree 4 [i]. Thus, it is natural to look for efficient approximation algo ri thms. Let A be an approximation algorithm. Denote by VC A the vertex cover which A produces for G, and by VC* a minimum vertex cover of G. Let E > O. We say that e is a performance bound of A if for every graph IVCA(-IVC*I fvc*l Gavril suggested a linear-time approximation algorithm for which e = 1 (see [i] page 134). Hochbaum [2] showed a polynomial-time (linear Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1982 ACM 0-89791-067-2/82/005/0303 \$00.75 programming) approximation algorithm for weighted graphs, for which e = I, and Bar-Yehuda and Even [3] achieved the same performance bound with a