Tropical Branching Programs

Pure DP algorithms use only the basic $$(\min ,+)$$ or $$(\max ,+)$$ operations in their recursion equations. Many of such algorithms, especially those for various shortest path problems, are even incremental in that one of the inputs to the addition gate is a variable. Just as tropical circuits constitute a natural mathematical model for pure DP algorithms, tropical branching programs constitute a natural mathematical model for incremental DP algorithms. In this chapter, we prove some general lower bounds for tropical branching programs and show that for some explicit optimization problems, non-incremental pure DP algorithms can be super-polynomially faster than incremental algorithms. This also implies that tropical formulas can be super-polynomially weaker than tropical circuits.