By Kim H.

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Thus,y E S n Co [=S,]and thereforey’E S’, [ C C g , 1. Case 2: x E Sk Thus, x E lo,x This proves (*). Now, w ( V , ) + 2 4 C , ) = w(V0 u = UCCB) c, u C’O) [by definitions of V O ,CO1 5 (cBl) =w ((V-S,) u S’C) = O(Vo u [by (*) and optimality of C B ~ c, u SI u S’,). = O(V,) + w ( C , ) + w (S+ +(S,). It follows that o(C,)

2) Using the Local-Ratio Corollary, with r which is a single edge. D. 4. Putting together NT and the local-ratio theorem Hochbaum [ 12 suggested the following approach to approximate WVC: Let C(V, El, w be the problem’s input, such that w(C*(C))2 1/2 d V ) . (This is achieved by the N T algorithm). Color C by k colors and let I be the “heaviest” monochromatic set of vertices. The cover produced is C= V-I. It follows that For general graphs she gets the ratio 2 degree) and for planar graphs - 2 - (A is the maximum A (k = 4) the performance ration 2 1 .

42 R Bar- Yehuda and S. Even though we get performance-ratios AE in linear time, we suspect that for any fixed AE, there is no polynomial time approximation algorithm with a better constant performance ratio, unless P=NP [even for the unweighted case], (this is an extension of a conjecture of Hochbaum). 6. Appendix - NT theorem Let G(V, El be a simple graph. Define the weights of vertices in I/’ by d u ’ ) = d u ) . Nemhauser and Trotter [ 181 presented the following local optiniz ation algorithm: Algorithm NT The following theorem states results of Nemhauser and Trotter, but our proof is shorter and does not use linear programming arguments.