FPT algorithms for a special block-structured integer program with applications in scheduling

IF 2.2 2区数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Mathematical Programming Pub Date : 2024-01-04 DOI:10.1007/s10107-023-02046-z

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Abstract

In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where \(B_i=B\) and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are interested in fixed parameter tractable (FPT) algorithms by taking as parameters the dimensions of the blocks and the largest coefficient. For almost combinatorial 4-block n-fold IPs, we first show that there exists some \(\lambda \le g(\gamma )\) such that for any nonzero kernel element \({\textbf{g}}\) , \(\lambda {\textbf{g}}\) can always be decomposed into kernel elements in the same orthant whose \(\ell _{\infty }\) -norm is bounded by \(g(\gamma )\) (while \({\textbf{g}}\) itself might not admit such a decomposition), where g is a computable function and \(\gamma \) is an upper bound on the dimensions of the blocks and the largest coefficient. Based on this, we are able to bound the \(\ell _{\infty }\) -norm of Graver basis elements by \({\mathcal {O}}(g(\gamma )n)\) and develop an \({\mathcal {O}}(g(\gamma )n^{3+o(1)}\hat{L}^2)\) -time algorithm (here \(\hat{L}\) denotes the logarithm of the largest absolute value occurring in the input). Additionally, we show that the \(\ell _{\infty }\) -norm of Graver basis elements is \(\varOmega (n)\) . As applications, almost combinatorial 4-block n-fold IPs can be used to model generalizations of classical problems, including scheduling with rejection, bi-criteria scheduling, and a generalized delivery problem. Therefore, our FPT algorithm establishes a general framework to settle these problems.

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一种特殊块结构整数程序的 FPT 算法及其在调度中的应用

摘要本文研究了广义 4 块 n 折 IP 的一个特例，其中 \(B_i=B\)且 B 的秩最多为 1。这种 IP 被称为近似组合 4 块 n 折 IP，包括广义 n 折 IP 的一个子例。我们感兴趣的是以块的维数和最大系数为参数的固定参数可操作性（FPT）算法。对于几乎是组合型的4块n折叠IP，我们首先证明存在一些 \(\lambda \le g(\gamma )\) 这样的内核元素：对于任何非零内核元素 \({\textbf{g}}\) 、 \(\lambda{textbf{g}}\)总是可以分解成同一个正交的内核元素，其\(\ell _{\infty }\) -norm受\(g(\gamma )\)约束（而\({textbf{g}}\)本身可能不允许这样的分解）、其中，g 是一个可计算的函数，而 \(\gamma \) 是块的维数和最大系数的上限。在此基础上，我们可以通过 \({\mathcal {O}}(g(\gamma )n)\) 来约束格拉弗基元的 \(ell _{\infty\ }) -norm，并开发出一种 \({\mathcal {O}}(g(\gamma )n^{3+o(1)}\hat{L}^2)\)-时间算法（这里的 \(hat{L}\ 表示输入中出现的最大绝对值的对数）。此外，我们还证明了 Graver 基元的 \(ell _\{infty }\) -norm是 \(\varOmega (n)\) 。作为应用，几乎可以用组合 4 块 n 折 IP 来模拟经典问题的一般化，包括拒绝调度、双标准调度和一般化交付问题。因此，我们的 FPT 算法建立了解决这些问题的通用框架。

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来源期刊

Mathematical Programming 数学-计算机：软件工程

CiteScore

5.70

自引率

11.10%

发文量

160

审稿时长

4-8 weeks

期刊介绍： Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.