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Functional Concurrent Regression Mixture Models Using Spiked Ewens-Pitman Attraction Priors. 使用尖刺Ewens-Pitman吸引先验的功能并发回归混合模型

IF 4.9 2区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Bayesian Analysis

Pub Date : 2024-12-01 Epub Date: 2023-05-02 DOI: 10.1214/23-ba1380

Mingrui Liang, Matthew D Koslovsky, Emily T Hébert, Michael S Businelle, Marina Vannucci

Functional concurrent, or varying-coefficient, regression models are a form of functional data analysis methods in which functional covariates and outcomes are collected concurrently. Two active areas of research for this class of models are identifying influential functional covariates and clustering their relations across observations. In various applications, researchers have applied and developed methods to address these objectives separately. However, no approach currently performs both tasks simultaneously. In this paper, we propose a fully Bayesian functional concurrent regression mixture model that simultaneously performs functional variable selection and clustering for subject-specific trajectories. Our approach introduces a novel spiked Ewens-Pitman attraction prior that identifies and clusters subjects' trajectories marginally for each functional covariate while using similarities in subjects' auxiliary covariate patterns to inform clustering allocation. Using simulated data, we evaluate the clustering, variable selection, and parameter estimation performance of our approach and compare its performance with alternative spiked processes. We then apply our method to functional data collected in a novel, smartphone-based smoking cessation intervention study to investigate individual-level dynamic relations between smoking behaviors and potential risk factors.

功能并发或变化系数回归模型是功能数据分析方法的一种形式，其中功能协变量和结果是同时收集的。这类模型的两个活跃研究领域是识别有影响的功能协变量和聚类它们在各观测值之间的关系。在各种应用中，研究人员已经应用和开发了一些方法来分别实现这些目标。然而，目前还没有一种方法能同时完成这两项任务。在本文中，我们提出了一种完全贝叶斯的功能并发回归混合模型，该模型可同时对特定受试者的轨迹进行功能变量选择和聚类。我们的方法引入了一种新颖的穗状 Ewens-Pitman 吸引先验，可识别和聚类每个功能协变量的受试者轨迹，同时利用受试者辅助协变量模式的相似性为聚类分配提供信息。通过模拟数据，我们评估了我们的方法在聚类、变量选择和参数估计方面的性能，并将其与其他尖峰过程进行了比较。然后，我们将我们的方法应用于一项基于智能手机的新型戒烟干预研究中收集的功能数据，以调查吸烟行为与潜在风险因素之间的个体水平动态关系。

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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

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

Adaptive quarklet tree approximation 自适应夸克树近似法

IF 1.7 3区数学 Q2 MATHEMATICS, APPLIED

Advances in Computational Mathematics

Pub Date : 2024-10-31 DOI: 10.1007/s10444-024-10205-9

Stephan Dahlke, Marc Hovemann, Thorsten Raasch, Dorian Vogel

This paper is concerned with near-optimal approximation of a given univariate function with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by hp-approximation techniques of Binev, we use the underlying tree structure of the frame elements to derive an adaptive algorithm that, under standard assumptions concerning the local errors, can be used to create approximations with an error close to the best tree approximation error for a given cardinality. We support our findings by numerical experiments demonstrating that this approach can be used to achieve inverse-exponential convergence rates.

本文涉及用多项式丰富小波框架（即所谓的夸克框架）的元素对给定的单变量函数进行近优逼近。受 Binev 的 hp 近似技术启发，我们利用框架元素的底层树形结构推导出一种自适应算法，在有关局部误差的标准假设下，该算法可用于创建误差接近给定心率的最佳树形近似误差的近似值。我们通过数值实验证明，这种方法可以达到反指数收敛率，从而支持我们的研究结果。

引用次数: 0

Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients 具有退化扩散和赫尔德连续系数的抛物线系统的部分梯度正则性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-31 DOI: 10.1016/j.na.2024.113691

We consider vector valued weak solutions

u : Ω_{T} \to R^{N}

with

N \in N

of degenerate or singular parabolic systems of type

\partial_{t} u - div a (z, u, D u) = 0 in Ω_{T} = Ω \times (0, T),

where

Ω

denotes an open set in

R^{n}

for

n \geq 1

and

T > 0

a finite time. Assuming that the vector field

a

is not of Uhlenbeck-type structure, satisfies

p

-growth assumptions and

(z, u) \mapsto a (z, u, ξ)

is Hölder continuous for every

ξ \in R^{N n}

, we show that the gradient

D u

is partially Hölder continuous, provided the vector field degenerates like that of the

p

-Laplacian for small gradients.

我们考虑矢量值弱解 u:ΩT→RN，N∈N 的∂tu-diva(z,u,Du)=0inΩT=Ω×(0,T)类型的退化或奇异抛物线系统，其中Ω表示 Rn 中的开集，n≥1，T>0 为有限时间。假定向量场 a 不是乌伦贝克型结构，满足 p 生长假设，且 (z,u)↦a(z,u,ξ) 对于每个 ξ∈RNn 都是霍尔德连续的，我们证明梯度 Du 部分是霍尔德连续的，条件是向量场像 p-Laplacian 的梯度一样退化为小梯度。

引用次数: 0

The Leray-Lions existence theorem under general growth conditions 一般增长条件下的勒雷-狮子存在定理

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-31 DOI: 10.1016/j.jde.2024.10.025

We prove an existence (and regularity) result of weak solutions

u \in W_{0}^{1, p} (Ω) \cap W_{loc}^{1, q} (Ω)

, to a Dirichlet problem for a second order elliptic equation in divergence form, under general and

p, q -

growth conditions of the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with

q = p

. We found a way to treat the general context with explicit dependence on

(x, u)

, other than on the gradient variable

ξ = D u

; these aspects require particular attention due to the

p, q

-context, with some differences and new difficulties compared to the standard case

p = q

我们证明了在微分算子的一般和 p,q 增长条件下，发散形式二阶椭圆方程的 Dirichlet 问题的弱解 u∈W01,p(Ω)∩Wloc1,q(Ω) 的存在性（和正则性）结果。这是首次尝试将众所周知的勒雷-狮子存在定理扩展到一般增长，该定理在 q=p 的所谓自然增长条件下成立。除了梯度变量ξ=Du之外，我们还找到了一种明确依赖 (x,u) 的一般情况下的处理方法；由于 p,q 条件，这些方面需要特别注意，与标准情况 p=q 相比，存在一些差异和新的困难。

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引用次数: 0

Extremal bounds for pattern avoidance in multidimensional 0-1 matrices 多维 0-1 矩阵中模式规避的极值界限

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114303

<div><div>A 0-1 matrix M contains another 0-1 matrix P if some submatrix of M can be turned into P by changing any number of 1-entries to 0-entries. The 0-1 matrix M is <math><mi>P</mi></math>-saturated where <math><mi>P</mi></math> is a family of 0-1 matrices if M avoids every element of <math><mi>P</mi></math> and changing any 0-entry of M to a 1-entry introduces a copy of some element of <math><mi>P</mi></math>. The extremal function <math><mi>ex</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math> and saturation function <math><mi>sat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math> are the maximum and minimum possible number of 1-entries in an <math><mi>n</mi><mo>×</mo><mi>n</mi></math> <math><mi>P</mi></math>-saturated 0-1 matrix, respectively, and the semisaturation function <math><mi>ssat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math> is the minimum possible number of 1-entries in an <math><mi>n</mi><mo>×</mo><mi>n</mi></math> <math><mi>P</mi></math>-semisaturated 0-1 matrix M, i.e., changing any 0-entry in M to a 1-entry introduces a new copy of some element of <math><mi>P</mi></math>.</div><div>We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-<math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math> d-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-<math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math> d-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers <math><mi>k</mi><mo>,</mo><mi>d</mi></math> and integer <math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math>, we construct a family of d-dimensional 0-1 matrices with both extremal function and saturation function exactly <math><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></math> for sufficiently large n. We show that no family of d-dimensional 0-1 matrices has saturation function strictly between <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> and <math><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math> and we construct a family of d-dimensional 0-1 matrices with bounded saturation function and extremal function <math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></spa

如果 M 的某个子矩阵可以通过将任意数量的 1 条目变为 0 条目而变成 P，则 0-1 矩阵 M 包含另一个 0-1 矩阵 P。如果 M 避开了 P 的每个元素，并且将 M 的任意 0 条目改为 1 条目都会引入 P 的某个元素的副本，那么 0-1 矩阵 M 就是 P 饱和的，其中 P 是 0-1 矩阵族。极值函数 ex(n,P) 和饱和函数 sat(n,P) 分别是 n×n P 饱和 0-1 矩阵中 1 条目的最大可能数目和最小可能数目，而半饱和函数 ssat(n,P) 是 n×n P 半饱和 0-1 矩阵 M 中 1 条目的最小可能数目，即、我们研究多维 0-1 矩阵的这些函数。特别是，我们给出了最小非 O（nd-1）d 维 0-1 矩阵参数的上限，这是从二维最小非线性 0-1 矩阵推广而来的；我们还证明了存在无限多的最小非 O（nd-1）d 维 0-1 矩阵，且所有维的长度都大于 1。对于任意正整数 k,d 和整数 r∈[0,d-1]，我们构造了一个 d 维 0-1 矩阵族，其极值函数和饱和函数在足够大的 n 条件下正好为 knr。我们证明没有一个 d 维 0-1 矩阵族的饱和函数严格介于 O(1) 和 Θ(n) 之间，并且我们构造了一个 d 维 0-1 矩阵族，其饱和函数和极值函数 Ω(nd-ϵ) 对于任意 ϵ>0 都是有界的。对于某个整数 r∈[0,d-1]，我们证明其半饱和函数总是 Θ(nr)。

引用次数: 0

Efficient constant-factor approximate enumeration of minimal subsets for monotone properties with weight constraints 带权重约束的单调属性最小子集的高效恒因子近似枚举

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.dam.2024.10.014

A property

Π

on a finite set

U

is monotone if for every

X \subseteq U

satisfying

Π

, every superset

Y \subseteq U

X

also satisfies

Π

. Many combinatorial properties can be seen as monotone properties. The problem of finding a subset of

U

satisfying

Π

with the minimum weight is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to enumerate multiple small solutions rather than to find a single small solution. To this end, given a weight function

w : U \to Q_{> 0}

and

k \in Q_{> 0}

, we devise algorithms that approximately enumerate all minimal subsets of

U

with weight at most

k

satisfying

Π

for various monotone properties

Π

, where “approximate enumeration” means that algorithms output all minimal subsets satisfying

Π

whose weight is at most

k

and may output some minimal subsets satisfying

Π

whose weight exceeds

k

but is at most

c k

for some constant

c \geq 1

. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most

k

with constant approximation factors.

如果对于满足 Π 的每个 X⊆U，X 的每个超集 Y⊆U 也满足 Π，那么有限集 U 上的属性 Π 就是单调的。许多组合性质都可以看作是单调性质。以最小权重找到满足 Π 的 U 子集是组合优化的核心问题。虽然已经开发出了许多近似/精确算法来解决这类问题，但由于很难针对实际问题建立精确的数学模型，这些算法得到的解决方案往往不适合实际应用。克服这一困难的一种可行方法是枚举多个小解决方案，而不是寻找单一的小解决方案。为此，给定一个权重函数 w:U→Q>0 和 k∈Q>0，我们设计了一些算法，可以近似枚举权重至多为 k、满足各种单调属性 Π 的 U 的所有最小子集，这里的 "近似枚举 "是指算法输出所有满足 Π 的最小子集，其权重至多为 k，并且可能输出一些满足 Π 的最小子集，其权重超过 k，但在某个常数 c≥1 时至多为 ck。通过这些算法，我们可以高效地枚举最小顶点覆盖、有界度图中的最小支配集、最小反馈顶点集、有界秩超图中的最小命中集等、权重最多为 k，且近似系数恒定。

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引用次数: 0

Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting 具有弱奇异敏感性和逻辑动力学的趋化系统在任意维度环境中的有界性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-31 DOI: 10.1016/j.jde.2024.10.027

<div><div>This paper deals with the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source(0.1)<math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>χ</mi><mi>∇</mi><mo>⋅</mo><mo>(</mo><mfrac><mrow><mi>u</mi></mrow><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>λ</mi></mrow></msup></mrow></mfrac><mi>∇</mi><mi>v</mi><mo>)</mo><mo>+</mo><mi>r</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>α</mi><mi>v</mi><mo>+</mo><mi>β</mi><mi>u</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>v</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math> where <math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>(</mo><mi>N</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math> is a smooth bounded domain, the parameters <math><mi>χ</mi><mo>,</mo><mspace></mspace><mi>r</mi><mo>,</mo><mspace></mspace><mi>μ</mi><mo>,</mo><mspace></mspace><mi>α</mi><mo>,</mo><mspace></mspace><mi>β</mi></math> are positive constants and <math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>.</div><div>It is well known that for parabolic-elliptic chemotaxis systems including singularity, a uniform-in-time positive pointwise lower bound for v is vitally important for establishing the global boundedness of classical solutions since the cross-diffusive term becomes unbounded near <math><mi>v</mi><mo>=</mo><mn>0</mn></math>. To this end, a key step in the literature is to establish a proper positive lower bound for the mass functional <math><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi></math>, which, due to the presence of logistic kinetics, is not preserved and hence it turns in for v. In contrast to this approach, in this article, the boundedness of classical solutions of (0.1) is obtained without using the uniformly positive lower bound of v.</div><div>Among others, it has been proven that without establishing a uniform-in-time positive pointwise lower bound for v, if <math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>, then there exists <math><mi>μ</mi><mo>></mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> such that for all suitably smooth in

本文讨论了以下抛物线-椭圆趋化竞争系统，该系统具有弱奇异敏感性和逻辑源（0.1){ut=Δu-χ∇-u(λv∇v)+ru-μu2,x∈Ω,0=Δv-αv+βu,x∈Ω,∂u∂ν=∂v∂ν=0∈∂Ω，其中Ω⊂RN(N≥1)为光滑有界域，参数χ,r,μ,αβ为正常数，λ∈(0,1)。众所周知，对于包含奇异性的抛物线-椭圆趋化系统，由于交叉扩散项在 v=0 附近变得无界，因此 v 的时间均匀正向点式下界对于确定经典解的全局有界性至关重要。为此，文献中的一个关键步骤是为质量函数∫ωu 建立适当的正下界，由于逻辑动力学的存在，质量函数∫ωu 是不保留的，因此它在 v 时会变为有界。与这种方法不同，本文在不使用 v 的均匀正下界的情况下得到了 (0.1) 经典解的有界性。其中，本文证明了在不建立 v 的时间均匀正点下限的情况下，若 λ∈(0,1)，则存在 μ>μ⁎，从而对于所有适当光滑的初始数据，任何全局定义正解的 Lp-norm（对于任意 p≥2）都是有界的；此外，问题（0.1）具有唯一的全局定义经典解。此外，在附加假设λ<12+1NwithN≥2 条件下，解也证明是均匀有界的。

引用次数: 0

Quasi-kernels in split graphs 分裂图中的准核

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.dam.2024.10.009

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The “small quasi-kernel conjecture”, proposed by Erdős and Székely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a

1 / 2

ratio, but even with larger ratio, this property is known to hold only for few classes of graphs.

The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph

D

has a quasi-kernel of size at most

\frac{2}{3} | V (D) |

, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.

在一个数图中，准核是一个独立的顶点子集，使得从每个顶点到这个子集的最短路径的长度最多为 2。Erdős 和 Székely 于 1976 年提出了 "小准核猜想"，假设每个无汇数图都有一个准核，其大小在顶点总数的几分之一以内。这一猜想在比率为 1/2 时更为精确，但即使比率更大，这一性质也只在少数几类图中成立。这个图族在该猜想的研究中发挥了特殊作用，因为它被用来推翻假设存在两个不相交准核的强化。本文证明了每个无汇分裂数图 D 最多有一个大小为 23|V(D)| 的准核，当该图是一个完整分裂图的定向时，甚至最多有一个大小为两个的准核。同时还证明，计算一个分裂数图中最小尺寸的准核是 W[2]-hard 的。

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引用次数: 0

Intersection of chordal graphs and some related partition problems 弦图的相交和一些相关的分割问题

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.dam.2024.10.010

The chordality of a graph is the minimum number of chordal graphs whose intersection is the graph. A result of Yannakakis’ from 1982 can be used to infer that for every fixed

k \geq 3

, deciding whether the chordality of a graph is at most

k

is NP-complete. We consider the problem of deciding whether the chordality of a graph is 2, or equivalently, deciding whether a given graph is the intersection of two chordal graphs. We prove that the problem is equivalent to a partition problem when one of the chordal graphs is a split graph and the other meets certain conditions. Using this we derive complexity results for a variety of problems, including deciding if a graph is the intersection of

k

split graphs, which is in P for

k = 2

and NP-complete for

k \geq 3

一个图的和弦度是其交集为该图的和弦图的最小数目。可以利用扬纳卡基斯（Yannakakis）1982 年的一个结果来推断，对于每个固定的 k≥3，判断一个图的和弦度是否最多为 k 是 NP-complete。我们考虑的问题是判定一个图的和弦度是否为 2，或者等价于判定一个给定的图是否是两个和弦图的交集。我们证明，当其中一个弦图是分裂图，而另一个满足特定条件时，该问题等同于分割问题。利用这一点，我们推导出了各种问题的复杂性结果，包括判定一个图是否是 k 个分裂图的交集，对于 k=2 的问题，该问题在 P 级，而对于 k≥3 的问题，该问题在 NP 级。

引用次数: 0

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