Network Flow
    exist vertex vs,t Uf(i,v)=Uf(v,j)
                                          i            j
 
    value of f=|f|=Uf(i,t)=Uf(s,j)
                              i        j
                 
    cut(x,x)Gx=v-x
          
      c(xx)=Uc(v,w)Gcut capacity
            v,w
             
    f(xx)=Uf(v,w)Gflow across the cut
            v,w
 
 
                                _
                         x      x          |f| = 14
                                                          
      where v belongs to x, w belongs to x
                                  
    LemmaG|f|=f(xx)-f(xx)c(xx)              
    pfG|f|=|f|+0+0=Uout(v)-Uin(v)=f(xx)-f(xx)
                   s,a,b        s,a,b     maxflow|f|=c(x,x)          12,5    8,3    7,2                 c,f
        SXaXbXt  +5,  Xf(e)<c(e)
               +7      +5     +5

               12,5   8,3     7,2                c,f
            SXaXbXt   +3,  Xf(e)>0
           +7     -3      +5

    find possible augmenting path, increase flow
 
                            4      10      5       12      7
s ----> c ----> d ----> a ----> b ----> t    +4
                           15      8       3       6       10
s ----> a ----> b ----> c ----> d ----> t    +3
                                    12      5       3   
s ----> a ----> b ----> t    +4
                                   9       -4      7 
s ----> a ----> d ----> t    +4
 
F = 14
      Irrational Capacity: might never terminate
    Even Integral capacity: 2M increments