 We consider the competitiveness of online algorithms in the random order input model (ROM) , when the offline vertices are weighted .
<meaning-changed>  We consider the competitiveness of online algorithms in the random order input model (ROM) , when the offline vertices are weighted .
 We consider the competitiveness of online algorithms in both the adversarial order model (AOM) and the random order model (ROM) , when the offline vertices are weighted .