Near-Linear Time Algorithm for $n$-Fold ILPs via Color Coding

Abstract

We study an important case of integer linear programs (ILPs) of the form $\max\{c^Tx \ \vert\ \mathcal Ax = b, l \leq x \leq u,\, x \in \mathbb{Z}^{n t} \} $ with $n t$ variables and lower and upper bounds $\ell, u\in\mathbb Z^{nt}$. In $n$-fold ILPs nonzero entries only appear in the first $r$ rows of the matrix $\mathcal A$ and in small blocks of size $s\times t$ along the diagonal underneath. Despite this restriction, many optimization problems can be expressed in this form. It is known that $n$-fold ILPs are fixed-parameter tractable (FPT) regarding the parameters $s, r,$ and $\Delta$, where $\Delta$ is the greatest absolute value of any entry in $\mathcal A$. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction where the number of iterations and the search for such an improving direction each take time $\Omega(n)$. This leads to a running time quadratic in $n$. We introduce a technique based on color coding which allows us to compute these improving directions in logarithmic time after a single initialization step. This yields an algorithm for $n$-fold ILPs with a running time that is near-linear in $nt$, the number of variables. More precisely, our algorithm runs in time $(rs\Delta)^{\mathcal{O}(r^2s + s^2)} L^2 nt \log^{\mathcal{O}(1)}(nt)$, where $L$ is the encoding length of the largest integer in the input. Further, in contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead we give a structural lemma to introduce appropriate bounds. On the other hand, if we are given such an LP solution, the running time can be decreased by a factor of $L$.