1、最小费用最大流问题 D=V A C(vi,vj)A b(vi,vj)0(bij).f b(f)=bijfij f D,vs vt=1 f f v(f)=v(f)+1 b(f)b(f):f v(f)*f*f f*v(f*)f*vs vt*f=0 0 f=0 f f W(f)D D(vi,vj)vi,vj(vj,vi)wij 1.vi,vj A D f W(f)vs vt 2.vj vi D vi vj 1 f(0)=0 2 k-1 f(k-1)W(f(k-1)3 W(f(k-1)vs vt f(k-1)(4)4 f(k 1)f(k 1)f(k)f(k)2 例:求图所示网络中的最小费用 最大流，弧
2、旁 的权是（bij,cij）.(bij,Cij)(1,8)vtv3 v2vsv1(3,10)(2,4)(6,2)(1,7)(4,10)(2,5)1 f(0)=0,W(f(0),vs vt(vs,v2,v1,vt)(1)vtv3 v2vsv1(3)(2)(6)(1)(4)(2)W(f(0)2 D=vs,v2,v1,vt(8)vtv3v2vsv1(10)(4)(2)(7)(10)(5)(5)vtv3 v2vsv1(0)(0)(0)(5)(0)(5)f(1),v(f(1)=53 f(0)=0=5 f(1)(1)vtv3 v2vsv1(3)(2)(6)(1)(4)(-2)W(f(1)(-1)(-1)4
3、)f(1)=5,W(f(1),vs vt(vs,v1,vt)(5)vtv3 v2vsv1(0)(0)(0)(5)(0)(5)f(1),v(f(1)=5(1)vtv3 v2vsv1(3)(2)(6)(1)(4)(-2)W(f(1)(-1)(-1)3）在 上对f(1)=5 进行 调整，(8)vtv3 v2vsv1(10)(4)(2)(7)(10)(5)(5)vtv3 v2vsv1(0)(0)(0)(5)(0)(5)(5)vtv3 v2vsv1(0)(0)(7)(2)(5)(0)f(2),v(f(2)=7=2，得到新可行流f(2)(5)vtv3v2vsv1(0)(0)(7)(2)(5)(0)f(2)
4、,v(f(2)=7(8)vtv3 v2vsv1(10)(4)(2)(7)(10)(5)(1)vtv3 v2vsv1(3)(2)(6)(-1)(4)(-2)(-1)(-4)W(f(2)(3)(-1)vtv3 v2vsv1(2)(6)(-1)(4)(-2)(-4)W(f(3)(-2)(-3)(8)vtv3 v2vsv1(3)(3)(0)(7)(2)(5)f(3),v(f(3)=10(8)vtv3 v2vsv1(10)(4)(2)(7)(10)(5)(8)vtv3 v2vsv1(4)(4)(0)(7)(3)(4)f(4),v(f(4)=11(8)vtv3 v2vsv1(10)(4)(2)(7)(10)(5)(-4)(-1)vtv3 v2v1(-2)(6)(-1)(4)(2)W(f(4)(-2)(-3)vs
