Problem $3.$ You are a wedding planner in charge of the seating chart for a wedding reception.

The guest list includes $n$ families, where family i has $a_i$ adult members. $($Children will sit at a

kids' table.$)$ The reception will take place in a ballroom with $m$ tables of various sizes, where

table $j$ can seat $b_j$ people. As any good wedding planner, you want a seating plan that does not

put two members of the same family at the same table, to stimulate mingling and conversation.

Formulate the problem of finding such a seating plan as a maximum flow model. Explain the list

of nodes in the network, and each arc's capacity. Explain how you would determine that no such

seating plan exists.