Discrete Optimization最新文献

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A polynomial-time algorithm for conformable coloring on regular bipartite and subcubic graphs 正则二部图和次三次图上的合着色的多项式时间算法

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-11-29 DOI: 10.1016/j.disopt.2024.100865

Luerbio Faria, Mauro Nigro, Diana Sasaki

In 1988, Chetwynd and Hilton observed that a

(Δ + 1)

-total coloring induces a vertex coloring in the graph, they called it conformable. A

(Δ + 1)

-vertex coloring of a graph

G = (V, E)

is called conformable if the number of color classes of parity different from that of

| V |

is at most the deficiency

def (G) = \sum_{v \in V} (Δ - d_{G} (v))

G

, where

d_{G} (v)

is the degree of a vertex

v

V

. In 1994, McDiarmid and Sánchez-Arroyo proved that deciding whether a graph

G

has

(Δ + 1)

-total coloring is NP-complete even when

G

k

-regular bipartite with

k \geq 3

. However, the time-complexity of the problem of determining whether a graph admits a conformable coloring (Conformability problem) remains unknown. In this paper, we prove that Conformability problem is polynomial solvable for the class of

k

-regular bipartite and for the class of subcubic graphs.

在1988年，Chetwynd和Hilton观察到（Δ+1）-total着色在图中引起顶点着色，他们称之为顺应着色。图G=（V，E）的A （Δ+1）-顶点着色，如果与|V|的奇偶性不同的色类数最多等于G的缺陷def(G)=∑V∈V（Δ−dG(V)），其中dG(V)是V的顶点V的度。1994年，McDiarmid和Sánchez-Arroyo证明了判定图G是否具有（Δ+1）-全着色是np完全的，即使G是k≥3的k正则二部。然而，确定图是否允许符合着色问题（符合问题）的时间复杂度仍然是未知的。本文证明了k正则二部图和次三次图的一致性问题是多项式可解的。

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引用次数: 0

Generalized min-up/min-down polytopes 广义最小上/最小下多面体

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-11-28 DOI: 10.1016/j.disopt.2024.100866

Cécile Rottner

Consider a time horizon and a set of

N

possible states for a given system. The system must be in exactly one state at a time. In this paper, we generalize classical results on min-up/min-down constraints for a 2-state system to an

N

-state system with

N \geq 3

. The minimum-time constraints enforce that if the system switches to state

i

at time

t

, then it must remain in state

i

for a minimum number of time steps. The minimum-time polytope is defined as the convex hull of integer solutions satisfying the minimum-time constraints. A variant of minimum-time constraints is also considered, namely the no-spike constraints. They enforce that if state

i

is switched on at time

t

, the system must remain on states

j \geq i

during a minimum time. Symmetrically, they also enforce that if state

i

is switched off at time

t

, the system must remain on states

j < i

during a minimum time. The no-spike polytope is defined as the convex hull of integer solutions satisfying the no-spike constraints. For both the minimum-time polytope and the no-spike polytope, we introduce families of valid inequalities. We prove that these inequalities are facet-defining and lead to a complete description of polynomial size for each polytope.

考虑一个给定系统的时间范围和一组N种可能状态。系统每次只能处于一种状态。在本文中，我们将经典的关于2态系统最小上/最小下约束的结果推广到N≥3的N态系统。最小时间约束强制要求，如果系统在时间t切换到状态i，那么它必须保持状态i的最小时间步长。最小时间多面体定义为满足最小时间约束的整数解的凸包。还考虑了最小时间约束的一种变体，即无尖峰约束。它们强制要求如果状态i在时间t开启，系统必须在最小时间内保持状态j≥i。对称地，它们还强制执行，如果状态i在时间t关闭，则系统必须在最小时间内保持状态j<；i。无尖峰多面体定义为满足无尖峰约束的整数解的凸包。对于最小时间多面体和无尖峰多面体，我们引入了有效不等式族。我们证明了这些不等式是面定义的，并给出了每个多面体多项式大小的完整描述。

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引用次数: 0

Corrigendum to “Bilevel time minimizing transportation problem” [Discrete Optim.] 5 (4) (2008) 714–723 双层时间最小化运输问题 "更正 [Discrete Optim.] 5 (4) (2008) 714-723

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-11-01 DOI: 10.1016/j.disopt.2024.100863

Sonia , Ankit Khandelwal

This is a corrigendum to our research paper titled “Bilevel time minimizing transportation problem” published in 2008. We deeply regret a minor error in the formulation of an intermediate problem solved as part of the algorithm. The intermediate problem,

{(T P)}_{t}^{T}

, used to iteratively generate the prospective solution pairs, was initially modeled as a linear programming problem. But the correct formulation of its objective function now involves a binary function, thus making it an NP-hard problem. The algorithm is no longer polynomially bound as it involves solving a finite number of mixed 0-1 programming problems. The manuscript’s original contribution stands correct and there is no change to the structure or the accuracy of the algorithm. The changes required to the original paper, due to this error, are presented in this corrigendum.

这是对我们 2008 年发表的题为 "双层时间最小化运输问题 "的研究论文的更正。我们对作为算法一部分的中间问题的表述中的一个小错误深表遗憾。用于迭代生成预期解对的中间问题 (TP)tT 最初被建模为线性规划问题。但其目标函数的正确表述现在涉及二元函数，从而使其成为一个 NP-困难问题。该算法不再具有多项式约束，因为它涉及求解有限数量的混合 0-1 编程问题。手稿的原始贡献是正确的，算法的结构和准确性没有改变。本更正介绍了因这一错误而需要对原论文进行的修改。

引用次数: 0

Anchor-robust project scheduling with non-availability periods 具有不可用时段的锚固型项目进度安排

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-11-01 DOI: 10.1016/j.disopt.2024.100864

Pascale Bendotti , Luca Brunod Indrigo , Philippe Chrétienne , Bruno Escoffier

In large-scale scheduling applications, it is often decisive to find reliable schedules prior to the execution of the project. Most of the time however, data is affected by various sources of uncertainty. Robust optimization is used to overcome this imperfect knowledge. Anchor robustness, as introduced in the literature for processing time uncertainty, makes it possible to guarantee job starting times for a subset of jobs. In this paper, anchor robustness is extended to the case where uncertain non-availability periods must be taken into account. Three problems are considered in the case of budgeted uncertainty: checking that a given subset of jobs is anchored in a given schedule, finding a schedule of minimal makespan in which a given subset of jobs is anchored and finding an anchored subset of maximum weight in a given schedule. Polynomial time algorithms are proposed for the first two problems while an inapproximability result is given for the third one.

在大规模日程安排应用中，在项目执行前找到可靠的日程安排往往具有决定性意义。然而，在大多数情况下，数据会受到各种不确定因素的影响。稳健优化就是用来克服这种不完善的知识。文献中针对处理时间不确定性提出的锚稳健性，可以保证工作子集的工作开始时间。本文将锚稳健性扩展到必须考虑不确定不可用时段的情况。在预算不确定的情况下，本文考虑了三个问题：检查给定计划中是否锚定了给定的工作子集；找到一个最小有效期的计划，其中锚定了给定的工作子集；找到给定计划中权重最大的锚定子集。针对前两个问题提出了多项式时间算法，针对第三个问题给出了不可逼近性结果。

引用次数: 0

Circuit and Graver walks and linear and integer programming 电路和格拉夫行走以及线性和整数编程

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-10-01 DOI: 10.1016/j.disopt.2024.100862

Shmuel Onn

We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.

我们证明，只需使用线性代数运算和单个给定线性规划的解，就能在多项式时间内计算从给定线性规划的给定可行点到最优点的电路行走。我们还证明，从给定整数程序的给定可行点到最优点的 Graver 走法可以用整数编程神谕在多项式时间内计算，但如果没有这样的神谕，即使给定程序的最优解也很难计算这样的走法。结合我们的神谕算法和稀疏整数编程的最新成果，我们还证明了在有界树深度和子决定子矩阵上，从任意点出发的格拉夫行走都是多项式时间可计算的。

引用次数: 0

Approximation schemes for Min-Sum k-Clustering 最小和 k 聚类的近似方案

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-09-17 DOI: 10.1016/j.disopt.2024.100860

Ismail Naderi, Mohsen Rezapour, Mohammad R. Salavatipour

We consider the Min-Sum $k$ -Clustering ( $k$ -MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph $G = (V, E)$ and a parameter $k$ , the goal is to partition the points $V$ into $k$ clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.

The $k$ -MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of $O (log | V |)$ in polynomial time for general metrics and an approximation ratio $2 + ϵ$ in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for $k$ -MSC (when $k$ is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from $k$ -MSC to the balanced $k$ -median problem and design approximation algorithms for the latter to obtain an approximation for $k$ -MSC.

In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for $k$ -MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced $k$ -median and is a trade off between center-based clustering problems (such as balanced $k$ -median) and pair-wise clustering (such as Min-Sum $k$ -clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.

我们考虑的是最小和 k 聚类（k-MSC）问题。给定一个度量中的点集（由边加权图 G=(V,E) 表示）和一个参数 k，目标是将点 V 划分为 k 个聚类，使得同一聚类中所有点对之间的距离之和最小。Behsaz 等人（2019）针对该问题获得的已知最佳近似算法在一般度量条件下的多项式时间内达到了 O(log|V|)的近似率，在具有约束倍维度的度量条件下的准多项式时间内达到了 2+ϵ 的近似率。在我们的研究之前，还没有针对任何非三维度量的 k-MSC 近似方案（当 k 是输入的一部分时）。事实上，之前的大部分研究都依赖于一个简单的事实，即从 k-MSC 到平衡 k-median 问题有一个 2 近似的还原，并为后者设计近似算法，从而得到 k-MSC 的近似值。在本文中，我们首次获得了该问题的准多项式时间近似方案（QPTAS），该方案适用于有界树宽、有界公路维数、有界倍维数（包括固定维数欧几里得度量）的图以及平面图和无次要图所诱导的度量。我们通过引入一个新的聚类问题（我们称之为 min-hub 聚类），绕过了 k-MSC 的 2 的障碍，它是平衡 k-median 的广义化，是基于中心的聚类问题（如平衡 k-median）和成对聚类问题（如 Min-Sum k-clustering）之间的权衡。然后，我们展示了如何在某些度量类别上找到 Min-hub 聚类的近似方案。

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引用次数: 0

Mostar index and bounded maximum degree 莫斯塔尔指数和有界最大度

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-09-14 DOI: 10.1016/j.disopt.2024.100861

Michael A. Henning , Johannes Pardey , Dieter Rautenbach , Florian Werner

<div>Došlić et al. defined the Mostar index of a graph <math><mi>G</mi></math> as <math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder><mrow><mo>|</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math>, where, for an edge <math><mrow><mi>u</mi><mi>v</mi></mrow></math> of <math><mi>G</mi></math>, the term <math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math> denotes the number of vertices of <math><mi>G</mi></math> that have a smaller distance in <math><mi>G</mi></math> to <math><mi>u</mi></math> than to <math><mi>v</mi></math>. For a graph <math><mi>G</mi></math> of order <math><mi>n</mi></math> and maximum degree at most <math><mi>Δ</mi></math>, we show <math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where <math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math> only depends on <math><mi>Δ</mi></math> and the <math><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math> term only depends on <math><mi>n</mi></math>. Furthermore, for integers <math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math> and <math><mi>Δ</mi></math> at least 3, we show the existence of a <math><mi>Δ</mi></math>-regular graph of order <math><mi>n</mi></math> at least <math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math> with <math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo></mrow></math> where <mat

Došlić 等人将图 G 的莫斯塔尔指数定义为 Mo(G)=∑uv∈E(G)|nG(u,v)-nG(v,u)| 其中，对于 G 的边 uv，nG(u,v) 表示 G 中与 u 的距离小于与 v 的距离的顶点数。对于阶数为 n、最大度数最多为 Δ 的图 G，我们证明了 Mo(G)≤Δ2n2-(1-o(1))cΔnlog(log(n)) ，其中 cΔ>0 只取决于 Δ，而 o(1) 项只取决于 n。此外，对于 n0 和 Δ 至少为 3 的整数，我们证明存在阶数至少为 n0 的 Δ 不规则图，其 Mo(G)≥Δ2n2-cΔ′nlog(n) ，其中 cΔ′>0 只取决于 Δ。

引用次数: 0

Easy and hard separation of sparse and dense odd-set constraints in matching 匹配中稀疏和密集奇集约束的易难分离

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-09-14 DOI: 10.1016/j.disopt.2024.100849

Brady Hunsaker, Craig Tovey

We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given size (e.g. $\leq 9 + | V | / 6$ ) are present. Our algorithm also solves the more general problem of finding a T-cut subject to upper bounds on the cardinality of its defining node set and on its cost. In contrast, regarding dense constraints, we prove that for every $0 < α \leq \frac{1}{2}$ , it is NP-complete to separate over the class of constraints on odd sets of size $2 ⌊ (1 + α | V |) / 2 ⌋ - 1$ or $\geq α | V |$ .

我们研究了介于分数匹配和完全匹配多边形之间的多边形，方法是施加严格的奇数集（开花）约束子集。对于稀疏约束，如果只存在以给定大小（如 ≤9+|V|/6）为边界的所有奇数集的约束，我们会给出一种多项式时间分离算法。我们的算法还能解决更普遍的问题，即根据定义节点集的万有引力和成本的上限找到 T 切。相反，关于密集约束，我们证明了对于每一个 0<α≤12，在大小为 2⌊(1+α|V|)/2⌋-1或≥α|V|的奇数集合上分离一类约束是 NP-完全的。

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引用次数: 0

Two-set inequalities for the binary knapsack polyhedra 二元节包多面体的两组不等式

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-09-12 DOI: 10.1016/j.disopt.2024.100859

Todd Easton , Jennifer Tryon , Fabio Vitor

This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.

本文介绍了两组不等式，这是一类适用于knapsack和多重knapsack问题的有效不等式。两组不等式由来自一个背包约束的两个任意变量集生成。这一类切割平面不是传统的提升类型，因为开始时并不需要限制空间上的有效不等式。此外，它们无法使用任何现有的提升技术进行推导。本文提出了一种四元算法，可以高效地生成许多二集不等式。同时还推导出了面定义二集不等式的条件。计算实验测试了这些不等式作为预处理切分与 CPLEX（一种高性能数学编程求解器）在默认设置下的对比。总体而言，双集不等式将解决某些基准多重背包实例的时间缩短了 80%。计算结果还显示了这一新型切割平面在解决具有计算挑战性的二进制整数程序方面的潜力。

引用次数: 0

Revisiting some classical linearizations of the quadratic binary optimization problem and linkages with constraint aggregations 重新审视二次二元优化问题的一些经典线性化方法以及与约束聚合的联系

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2024-08-19 DOI: 10.1016/j.disopt.2024.100858

Abraham P. Punnen, Navpreet Kaur Dhanda

In this paper, we examine explicit linearization models associated with the quadratic unconstrained binary optimization problem (QUBO). After reviewing some well-known basic linearization techniques, we present a new explicit linearization technique (PK) for QUBO with interesting properties. In particular, PK yields the same LP relaxation value as that of the standard linearization of QUBO while employing less number of constraints. We then show that using weighted aggregations of selected constraints of the basic linearization models, several new and effective linearization models of QUBO can be developed. Although aggregation of constraints has been studied in the past for solving systems of Diophantine equations in non-negative variables, none of them resulted in practical models, particularly due to the large size of the associated multipliers. For our aggregation based models, the multipliers can be of any positive real number. Moreover, we show that by choosing the multipliers appropriately, the LP relaxations of the resulting linearizations have optimal objective function values identical to that of the corresponding non-aggregated models. Theoretical and experimental comparisons of the new and existing explicit linearization models are also provided. Although our discussions were focused primarily on QUBO, the models and results we obtained extend naturally to the quadratic binary optimization problem with linear constraints.

本文研究了与二次无约束二元优化问题（QUBO）相关的显式线性化模型。在回顾了一些著名的基本线性化技术后，我们提出了一种新的 QUBO 显式线性化技术（PK），它具有有趣的特性。特别是，PK 与 QUBO 标准线性化的 LP 松弛值相同，但采用的约束条件较少。然后我们证明，利用基本线性化模型中选定约束条件的加权聚合，可以开发出几种新的、有效的 QUBO 线性化模型。尽管过去曾有人研究过利用约束条件的聚合来求解非负变量的二叉方程组，但没有一个模型是实用的，特别是由于相关乘数的规模太大。对于我们基于聚合的模型，乘数可以是任何正实数。此外，我们还证明，通过适当选择乘数，所得到的线性化的 LP 放松具有与相应的非聚合模型相同的最佳目标函数值。我们还对新的和现有的显式线性化模型进行了理论和实验比较。虽然我们的讨论主要集中在 QUBO 上，但我们所获得的模型和结果可以自然地扩展到具有线性约束的二次二元优化问题。

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