Linear Programming

Maximum flow is a cool problem, but I'm told that it's not used all that often in practice because most real-world problems are more complicated. Instead, a really big hammer is used: linear programming (LP).

More general than matching or network flow: , maximize . Here, A is a matrix, x a column vector of variables, b a column vectors of constraints, and c a column vector of rewards.

Network-Flow example:

• One variable for each edge, constrained to be between zero (no flow) and the capacity (biggest possible flow).
• Constraint for each node: sum of value of variables on incoming edges must equal the sum of value of variables on outgoing edges (flow conservation).
• Maximize sum of edges leaving the source (size of the flow).

Small example...