Under Construction
(PVC) MINIMUM PARTIAL VERTEX COVER
Instance: Graph G=(V,E), for each vertex v in V a weight W(v)>=0;
          a number t>0;
Solution: A t-vertex cover for G, i.e., a subset C of V such that,
          at lease t edges are covered by C ( |E(C)| >= t)
Measure : W(C) (i.e. Sum {W(v): v in C})

+------------------------------------------------------------------+
|   The "Local-Ratio Theorem" (oriented for PVC)                   |
+------------------------------------------------------------------+
|   If   C is 2-approximation t-VC w.r.t the weight function W1    |
|   and  C is 2-approximation t-VC w.r.t the weight function W2    |
|   then C is 2-approximation t-VC w.r.t the weight function W1+W2 |
+------------------------------------------------------------------+

Our objective is to find a MINIMAL-t-VC C, i.e.
C is an t-VC, but for each v in C: C-{v} is not t-VC.

Define W1 to be epsilon HOMOGENEOUS, i.e.
for all v in V: W1(v) = epsilon*Min(t,degree(v))

Q1: Every MINIMAL-FVS is 2-approximations w.r.t W1. Why?
A1: See my paper from SODA99.

+-------------------------------------------------------------------+
| Algorithm Partial_VC (G=(V,E),W,t)                                |
+-------------------------------------------------------------------+
|    If |E|<t return "no solution"                                  |
|    If t<=0 is a tree return empty set                             |
|    If exists_{x in V} W(x)=0 return {x}+Partial_VC(G-{x},W,t-degree(x)
|    Let epsilon = min_{v in V} W(v)/Min(t,degree(v))               |
|    {The Local-Ratio step:}                                        |
|    Define W : forall_{x in V} W1(x) = epsilon*Min(t,degree(x))    |
|    C = Partial_VC(G,W-W1,t)                                       |
|    {"Make C Minimal" loop:}                                       |
|    for each x in C, if W1(x) > 0 and C-{x} is t-VC for G          |
|       then C=C-{x}                                                |
|    Return C                                                       |
+-------------------------------------------------------------------+

In the SODA99 paper, we generalize the problems so that
the graph may be hypergraph (see PVC: Partial Set-Cover) and each
edge have a "length" so we need length(E(C))>=t.