SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2282-2299, January 2020.
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 $L$.

中文翻译：

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通过颜色编码的$ n $折叠ILP的近线性时间算法

SIAM离散数学杂志，第34卷，第4期，第2282-2299页，2020年1月。
我们研究整数线性程序（ILP）的重要情况，其形式为$ \ max \ {c ^ Tx \ \ vert \ \ mathcal Ax = b，l \ leq x \ leq u，\，x \ in \ mathbb {Z } ^ {nt} \} $，带有$ nt $变量以及上下限$ \ ell，u \ in \ mathbb Z ^ {nt} $。在$ n $倍的ILP中，非零项仅出现在矩阵$ \ mathcal A $的前$ r $行中，以及在其对角线下方的小块$ s \ timest $中。尽管有此限制，但许多优化问题仍可以以此形式表示。众所周知，关于参数$ s，r，$和$ \ Delta $，$ n $ -fold ILP是固定参数易处理（FPT），其中$ \ Delta $是$ \ mathcal中任何条目的最大绝对值A $。最新技术是一种局部搜索算法，其随后沿改进方向移动，其中迭代次数和对该改进方向的搜索每个都花费时间$ \ Omega（n）$。这导致$ n $的运行时间为平方。我们介绍了一种基于颜色编码的技术，该技术使我们可以在单个初始化步骤后以对数时间计算这些改进方向。这产生了一个用于$ n $倍ILP的算法，其运行时间在变量数$ nt $中几乎是线性的。更准确地说，我们的算法在$（rs \ Delta）^ {\ mathcal {O}（r ^ 2s + s ^ 2）} L ^ 2 nt \ log ^ {\ mathcal {O}（1）}（nt ）$，其中$ L $是输入中最大整数的编码长度。此外，与最近文献中的算法相比，我们不需要解决LP松弛来处理无界变量。取而代之的是，我们给出结构上的引理来引入适当的界限。另一方面，如果给我们这样的LP解决方案，运行时间可以减少$ L $倍。