We study the general integer programming problem where the number of
variables n is a variable part of the input. We consider two natural
parameters of the constraint matrix A: its numeric measure a and its
sparsity measure d. We show that integer programming can be solved in time
g(a,d)poly(n,L), where g is some computable function of the
parameters a and d, and L is the binary encoding length of the input. In
particular, integer programming is fixed-parameter tractable parameterized by
a and d, and is solvable in polynomial time for every fixed a and d.
Our results also extend to nonlinear separable convex objective functions.
Moreover, for linear objectives, we derive a strongly-polynomial algorithm,
that is, with running time g(a,d)poly(n), independent of the rest of
the input data.
We obtain these results by developing an algorithmic framework based on the
idea of iterative augmentation: starting from an initial feasible solution, we
show how to quickly find augmenting steps which rapidly converge to an optimum.
A central notion in this framework is the Graver basis of the matrix A, which
constitutes a set of fundamental augmenting steps. The iterative augmentation
idea is then enhanced via the use of other techniques such as new and improved
bounds on the Graver basis, rapid solution of integer programs with bounded
variables, proximity theorems and a new proximity-scaling algorithm, the notion
of a reduced objective function, and others.
As a consequence of our work, we advance the state of the art of solving
block-structured integer programs. In particular, we develop near-linear time
algorithms for n-fold, tree-fold, and 2-stage stochastic integer programs.
We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified
proximity-scaling algorith