Parameterized Algorithms for MILPs with Small Treedepth

TL;DR

This paper introduces an algorithm for solving mixed integer linear programs efficiently when the constraint matrix has small treedepth, extending recent structural results from ILPs to the mixed case.

Contribution

The paper develops a new algorithm for MILPs with small treedepth, providing bounds on vertex denominators and analyzing the structure of the constraint matrix.

Findings

Algorithm runs in time f(a,d) * poly(n) for MILPs with small treedepth.

Bounded denominators of vertices enable scaling to integer grids.

Restricting structure only on integral variables does not ensure tractability.

Abstract

Solving (mixed) integer linear programs, (M)ILPs for short, is a fundamental optimization task. While hard in general, recent years have brought about vast progress for solving structurally restricted, (non-mixed) ILPs: $n$-fold, tree-fold, 2-stage stochastic and multi-stage stochastic programs admit efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth. In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time $f(a,d) \textrm{poly}(n)$, where $a$ is the largest coefficient of the constraint matrix, $d$ is its treedepth, and $n$ is the number of variables. This is enabled by proving bounds on the denominators of the vertices of bounded-treedepth (non-integer) linear programs. We do so by carefully analyzing the inverses of invertible submatrices of the constraint matrix. This allows us to afford scaling up the mixed program to the integer grid, and applying the known methods for integer programs. We trace the limiting boundary of our approach, showing that naturally related classes of linear programs have vertices of unbounded fractionality. Finally, we show that restricting the structure of only the integral variables in the constraint matrix does not yield tractable special cases.