Proof-techniques for the hardness of optimization problems (esp. Polynomial time) I've given an optimization problem for which I want to show that it is solvable in polynomial time.

Now, I have two questions:


Can this be done by formulating a mixed-integer linear program such that the coefficient matrix A has only values of -1, 0 or +1 and show that A is total unimodular? Because solving the relaxation of this problem with appropriate LP-techniques delivers an optimal solution in polynomial time.
What alternatives do I have to show that an optimization problem is solvable in polynomial time? I'm searching for an elegant proof for writing down this result.


EDIT: I reformulated my questions and hope my request becomes clearer.
Regarding question 2, there are at least two other classes of matrices that give integral polehydra: the balanced and totally balanced matrices. When available, these properties are simpler to establish than total unimodularity.

I'm not sure about question 1: your first paragraph seems to contain the answer, but after closer inspection I realize that total unimodularity yields integral polyhedras for arbitrary budget vectors. As Linear Programming is not known to be strongly polynomial, I suspect that you won't get polynomial time with an arbitrary integral budget that is binary-encoded (but it works say for a 0,1-budget vector).

You can find some relevant lecture notes online; some textbooks might be useful too (such as "Combinatorial Optimization" or "Hypergraph Theory: An Introduction").