Pub Date : 2024-12-01Epub Date: 2023-05-02DOI: 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.
{"title":"Functional Concurrent Regression Mixture Models Using Spiked Ewens-Pitman Attraction Priors.","authors":"Mingrui Liang, Matthew D Koslovsky, Emily T Hébert, Michael S Businelle, Marina Vannucci","doi":"10.1214/23-ba1380","DOIUrl":"10.1214/23-ba1380","url":null,"abstract":"<p><p>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.</p>","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11507269/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43506288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-01DOI: 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.
{"title":"Anchor-robust project scheduling with non-availability periods","authors":"","doi":"10.1016/j.disopt.2024.100864","DOIUrl":"10.1016/j.disopt.2024.100864","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 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。
{"title":"Adaptive quarklet tree approximation","authors":"Stephan Dahlke, Marc Hovemann, Thorsten Raasch, Dorian Vogel","doi":"10.1007/s10444-024-10205-9","DOIUrl":"10.1007/s10444-024-10205-9","url":null,"abstract":"<div><p>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 <i>hp</i>-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.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10205-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142555220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.na.2024.113691
We consider vector valued weak solutions with of degenerate or singular parabolic systems of type where denotes an open set in for and a finite time. Assuming that the vector field is not of Uhlenbeck-type structure, satisfies -growth assumptions and is Hölder continuous for every , we show that the gradient is partially Hölder continuous, provided the vector field degenerates like that of the -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。
{"title":"Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients","authors":"","doi":"10.1016/j.na.2024.113691","DOIUrl":"10.1016/j.na.2024.113691","url":null,"abstract":"<div><div>We consider vector valued weak solutions <span><math><mrow><mi>u</mi><mo>:</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi></mrow></math></span> of degenerate or singular parabolic systems of type <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>div</mi><mspace></mspace><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>Ω</mi></math></span> denotes an open set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span> a finite time. Assuming that the vector field <span><math><mi>a</mi></math></span> is not of Uhlenbeck-type structure, satisfies <span><math><mi>p</mi></math></span>-growth assumptions and <span><math><mrow><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>↦</mo><mi>a</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> is Hölder continuous for every <span><math><mrow><mi>ξ</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mi>n</mi></mrow></msup></mrow></math></span>, we show that the gradient <span><math><mrow><mi>D</mi><mi>u</mi></mrow></math></span> is partially Hölder continuous, provided the vector field degenerates like that of the <span><math><mi>p</mi></math></span>-Laplacian for small gradients.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.jde.2024.10.025
We prove an existence (and regularity) result of weak solutions , to a Dirichlet problem for a second order elliptic equation in divergence form, under general and 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 . We found a way to treat the general context with explicit dependence on , other than on the gradient variable ; these aspects require particular attention due to the -context, with some differences and new difficulties compared to the standard case .
{"title":"The Leray-Lions existence theorem under general growth conditions","authors":"","doi":"10.1016/j.jde.2024.10.025","DOIUrl":"10.1016/j.jde.2024.10.025","url":null,"abstract":"<div><div>We prove an existence (and regularity) result of weak solutions <span><math><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∩</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></math></span>, to a Dirichlet problem for a second order elliptic equation in divergence form, under general and <span><math><mi>p</mi><mo>,</mo><mi>q</mi><mo>−</mo></math></span><em>growth conditions</em> 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 <span><math><mi>q</mi><mo>=</mo><mi>p</mi></math></span>. We found a way to treat the general context with explicit dependence on <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span>, other than on the gradient variable <span><math><mi>ξ</mi><mo>=</mo><mi>D</mi><mi>u</mi></math></span>; these aspects require particular attention due to the <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span>-context, with some differences and new difficulties compared to the standard case <span><math><mi>p</mi><mo>=</mo><mi>q</mi></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.disc.2024.114303
<div><div>A 0-1 matrix <em>M</em> contains another 0-1 matrix <em>P</em> if some submatrix of <em>M</em> can be turned into <em>P</em> by changing any number of 1-entries to 0-entries. The 0-1 matrix <em>M</em> is <span><math><mi>P</mi></math></span>-saturated where <span><math><mi>P</mi></math></span> is a family of 0-1 matrices if <em>M</em> avoids every element of <span><math><mi>P</mi></math></span> and changing any 0-entry of <em>M</em> to a 1-entry introduces a copy of some element of <span><math><mi>P</mi></math></span>. The extremal function <span><math><mi>ex</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> and saturation function <span><math><mi>sat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> are the maximum and minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-saturated 0-1 matrix, respectively, and the semisaturation function <span><math><mi>ssat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> is the minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-semisaturated 0-1 matrix <em>M</em>, i.e., changing any 0-entry in <em>M</em> to a 1-entry introduces a new copy of some element of <span><math><mi>P</mi></math></span>.</div><div>We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-<span><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></span> <em>d</em>-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-<span><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></span> <em>d</em>-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers <span><math><mi>k</mi><mo>,</mo><mi>d</mi></math></span> and integer <span><math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, we construct a family of <em>d</em>-dimensional 0-1 matrices with both extremal function and saturation function exactly <span><math><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for sufficiently large <em>n</em>. We show that no family of <em>d</em>-dimensional 0-1 matrices has saturation function strictly between <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and we construct a family of <em>d</em>-dimensional 0-1 matrices with bounded saturation function and extremal function <span><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)。
{"title":"Extremal bounds for pattern avoidance in multidimensional 0-1 matrices","authors":"","doi":"10.1016/j.disc.2024.114303","DOIUrl":"10.1016/j.disc.2024.114303","url":null,"abstract":"<div><div>A 0-1 matrix <em>M</em> contains another 0-1 matrix <em>P</em> if some submatrix of <em>M</em> can be turned into <em>P</em> by changing any number of 1-entries to 0-entries. The 0-1 matrix <em>M</em> is <span><math><mi>P</mi></math></span>-saturated where <span><math><mi>P</mi></math></span> is a family of 0-1 matrices if <em>M</em> avoids every element of <span><math><mi>P</mi></math></span> and changing any 0-entry of <em>M</em> to a 1-entry introduces a copy of some element of <span><math><mi>P</mi></math></span>. The extremal function <span><math><mi>ex</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> and saturation function <span><math><mi>sat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> are the maximum and minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-saturated 0-1 matrix, respectively, and the semisaturation function <span><math><mi>ssat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> is the minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-semisaturated 0-1 matrix <em>M</em>, i.e., changing any 0-entry in <em>M</em> to a 1-entry introduces a new copy of some element of <span><math><mi>P</mi></math></span>.</div><div>We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-<span><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></span> <em>d</em>-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-<span><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></span> <em>d</em>-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers <span><math><mi>k</mi><mo>,</mo><mi>d</mi></math></span> and integer <span><math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, we construct a family of <em>d</em>-dimensional 0-1 matrices with both extremal function and saturation function exactly <span><math><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for sufficiently large <em>n</em>. We show that no family of <em>d</em>-dimensional 0-1 matrices has saturation function strictly between <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and we construct a family of <em>d</em>-dimensional 0-1 matrices with bounded saturation function and extremal function <span><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","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142555007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.dam.2024.10.014
A property on a finite set is monotone if for every satisfying , every superset of also satisfies . Many combinatorial properties can be seen as monotone properties. The problem of finding a subset of satisfying 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 and , we devise algorithms that approximately enumerate all minimal subsets of with weight at most satisfying for various monotone properties , where “approximate enumeration” means that algorithms output all minimal subsets satisfying whose weight is at most and may output some minimal subsets satisfying whose weight exceeds but is at most for some constant . 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 with constant approximation factors.
{"title":"Efficient constant-factor approximate enumeration of minimal subsets for monotone properties with weight constraints","authors":"","doi":"10.1016/j.dam.2024.10.014","DOIUrl":"10.1016/j.dam.2024.10.014","url":null,"abstract":"<div><div>A property <span><math><mi>Π</mi></math></span> on a finite set <span><math><mi>U</mi></math></span> is <em>monotone</em> if for every <span><math><mrow><mi>X</mi><mo>⊆</mo><mi>U</mi></mrow></math></span> satisfying <span><math><mi>Π</mi></math></span>, every superset <span><math><mrow><mi>Y</mi><mo>⊆</mo><mi>U</mi></mrow></math></span> of <span><math><mi>X</mi></math></span> also satisfies <span><math><mi>Π</mi></math></span>. Many combinatorial properties can be seen as monotone properties. The problem of finding a subset of <span><math><mi>U</mi></math></span> satisfying <span><math><mi>Π</mi></math></span> 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 <em>enumerate</em> multiple small solutions rather than to <em>find</em> a single small solution. To this end, given a weight function <span><math><mrow><mi>w</mi><mo>:</mo><mi>U</mi><mo>→</mo><msub><mrow><mi>Q</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>∈</mo><msub><mrow><mi>Q</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></mrow></math></span>, we devise algorithms that <em>approximately</em> enumerate all minimal subsets of <span><math><mi>U</mi></math></span> with weight at most <span><math><mi>k</mi></math></span> satisfying <span><math><mi>Π</mi></math></span> for various monotone properties <span><math><mi>Π</mi></math></span>, where “approximate enumeration” means that algorithms output all minimal subsets satisfying <span><math><mi>Π</mi></math></span> whose weight is at most <span><math><mi>k</mi></math></span> and may output some minimal subsets satisfying <span><math><mi>Π</mi></math></span> whose weight exceeds <span><math><mi>k</mi></math></span> but is at most <span><math><mrow><mi>c</mi><mi>k</mi></mrow></math></span> for some constant <span><math><mrow><mi>c</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. 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 <span><math><mi>k</mi></math></span> with constant approximation factors.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 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<span><span><span>(0.1)</span><span><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></span></span></span> where <span><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></span> is a smooth bounded domain, the parameters <span><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></span> are positive constants and <span><math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>.</div><div>It is well known that for parabolic-elliptic chemotaxis systems including singularity, a uniform-in-time positive pointwise lower bound for <em>v</em> is vitally important for establishing the global boundedness of classical solutions since the cross-diffusive term becomes unbounded near <span><math><mi>v</mi><mo>=</mo><mn>0</mn></math></span>. To this end, a key step in the literature is to establish a proper positive lower bound for the mass functional <span><math><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi></math></span>, which, due to the presence of logistic kinetics, is not preserved and hence it turns in for <em>v</em>. 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 <em>v</em>.</div><div>Among others, it has been proven that without establishing a uniform-in-time positive pointwise lower bound for <em>v</em>, if <span><math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, then there exists <span><math><mi>μ</mi><mo>></mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> 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。
{"title":"Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting","authors":"","doi":"10.1016/j.jde.2024.10.027","DOIUrl":"10.1016/j.jde.2024.10.027","url":null,"abstract":"<div><div>This paper deals with the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source<span><span><span>(0.1)</span><span><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></span></span></span> where <span><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></span> is a smooth bounded domain, the parameters <span><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></span> are positive constants and <span><math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>.</div><div>It is well known that for parabolic-elliptic chemotaxis systems including singularity, a uniform-in-time positive pointwise lower bound for <em>v</em> is vitally important for establishing the global boundedness of classical solutions since the cross-diffusive term becomes unbounded near <span><math><mi>v</mi><mo>=</mo><mn>0</mn></math></span>. To this end, a key step in the literature is to establish a proper positive lower bound for the mass functional <span><math><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi></math></span>, which, due to the presence of logistic kinetics, is not preserved and hence it turns in for <em>v</em>. 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 <em>v</em>.</div><div>Among others, it has been proven that without establishing a uniform-in-time positive pointwise lower bound for <em>v</em>, if <span><math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, then there exists <span><math><mi>μ</mi><mo>></mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> such that for all suitably smooth in","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-30DOI: 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 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 has a quasi-kernel of size at most , 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.
{"title":"Quasi-kernels in split graphs","authors":"","doi":"10.1016/j.dam.2024.10.009","DOIUrl":"10.1016/j.dam.2024.10.009","url":null,"abstract":"<div><div>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 <span><math><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span> ratio, but even with larger ratio, this property is known to hold only for few classes of graphs.</div><div>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 <span><math><mi>D</mi></math></span> has a quasi-kernel of size at most <span><math><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, 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.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-30DOI: 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 , deciding whether the chordality of a graph is at most 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 split graphs, which is in P for and NP-complete for .
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。
{"title":"Intersection of chordal graphs and some related partition problems","authors":"","doi":"10.1016/j.dam.2024.10.010","DOIUrl":"10.1016/j.dam.2024.10.010","url":null,"abstract":"<div><div>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 <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, deciding whether the chordality of a graph is at most <span><math><mi>k</mi></math></span> 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 <span><math><mi>k</mi></math></span> split graphs, which is in P for <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span> and NP-complete for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}