Discrete Applied Mathematics最新文献_第7页

The path-factors and generalized distance spectral radius of graphs 图的路径因子和广义距离谱半径

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-08 DOI: 10.1016/j.dam.2025.12.003

Yanhong Zhang , Lei Zhang , Haizhen Ren

Let

G

be a connected graph. A spanning subgraph

F

of

G

is called a path-factor if each component of

F

is a path. Let

k \geq 2

be an integer. A

P_{\geq k}

-factor of

G

is its spanning subgraph such that each component is a path of order at least

k

. The generalized distance matrix

D_{α} (G)

of

G

is defined as

D_{α} (G) = α T r (G) + (1 - α) D (G)

, where

0 \leq α \leq 1

. Inspired by the work of Cioabǎ et al. (2009), Suil (2021) and Zhou et al. (2024), we mainly study the existence of a

P_{\geq 2}

-factor in

G

based on the generalized distance spectral radius (

\partial (G)

) since the research on the path-factors is incomplete about the distance spectrum. We provide the lower bounds of the generalized distance spectral radius to guarantee the existence of a

P_{\geq 2}

-factor in a connected graph

G

with order

n \geq 4

. Furthermore, we verify a graph to show that the bound on generalized distance spectral radius is optimal. These improve the related existing results of Zhou et al. on the existence of a

P_{\geq 2}

-factor.

设G是连通图。如果生成子图F (G)的每个分量都是一条路径，则称之为路径因子。设k≥2为整数。G的P≥k因子是它的生成子图，使得每个分量都是至少k阶的路径。定义G的广义距离矩阵Dα(G)为Dα(G)=α tr (G)+(1−α)D(G)，其中0≤α≤1。由于距离谱的路径因子研究尚不完整，受cioabjii et al.（2009）、Suil（2021）和Zhou et al.（2024）等人工作的启发，我们主要基于广义距离谱半径（∂(G)）研究G中P≥2因子的存在性。在n≥4阶连通图G中，给出了广义距离谱半径的下界，以保证P≥2因子的存在。进一步，我们验证了一个图，证明了广义距离谱半径的界是最优的。这些改进了Zhou等人已有的关于P≥2因子存在性的相关结果。

{"title":"The path-factors and generalized distance spectral radius of graphs","authors":"Yanhong Zhang , Lei Zhang , Haizhen Ren","doi":"10.1016/j.dam.2025.12.003","DOIUrl":"10.1016/j.dam.2025.12.003","url":null,"abstract":"<div><div>Let <math><mi>G</mi></math> be a connected graph. A spanning subgraph <math><mi>F</mi></math> of <math><mi>G</mi></math> is called a path-factor if each component of <math><mi>F</mi></math> is a path. Let <math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math> be an integer. A <math><msub><mrow><mi>P</mi></mrow><mrow><mo>≥</mo><mi>k</mi></mrow></msub></math>-factor of <math><mi>G</mi></math> is its spanning subgraph such that each component is a path of order at least <math><mi>k</mi></math>. The generalized distance matrix <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> of <math><mi>G</mi></math> is defined as <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>T</mi><mi>r</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, where <math><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></math>. Inspired by the work of Cioabǎ et al. (2009), Suil (2021) and Zhou et al. (2024), we mainly study the existence of a <math><msub><mrow><mi>P</mi></mrow><mrow><mo>≥</mo><mn>2</mn></mrow></msub></math>-factor in <math><mi>G</mi></math> based on the generalized distance spectral radius (<math><mrow><mi>∂</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>) since the research on the path-factors is incomplete about the distance spectrum. We provide the lower bounds of the generalized distance spectral radius to guarantee the existence of a <math><msub><mrow><mi>P</mi></mrow><mrow><mo>≥</mo><mn>2</mn></mrow></msub></math>-factor in a connected graph <math><mi>G</mi></math> with order <math><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math>. Furthermore, we verify a graph to show that the bound on generalized distance spectral radius is optimal. These improve the related existing results of Zhou et al. on the existence of a <math><msub><mrow><mi>P</mi></mrow><mrow><mo>≥</mo><mn>2</mn></mrow></msub></math>-factor.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 234-240"},"PeriodicalIF":1.0,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737328","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}

引用次数: 0

Busy agents on a line 座席在线忙

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-05 DOI: 10.1016/j.dam.2025.11.049

Stefan Dobrev , Rastislav Královič , Dana Pardubská

In this paper we initiate the study of agent-based busy beaver problem: Consider a synchronous system consisting of an infinite discrete line and a group of

k

distinct (each with its own algorithm) agents, each being an

s

-state automaton that can see only the states (and IDs) of the co-located agents. Assuming that all agents start co-located, what is length of the longest terminating computation these agents can perform?

We give asymptotically optimal answer of

Θ (s^{3})

for

k = 2

, show that the answer is not a computable function for

k \geq 3

, give a deterministic construction for

k = 3

achieving at least

2^{2^{Ω (s^{0.347})}}

steps, and show that 3 agents can simulate a Turing machine. We also show that any

s

-state Turing machine can be simulated by a team of

O (log s)

oblivious agents.

As an application of our results, we show how to solve the following variant of the Treasure Hunt problem: There is no guarantee that the treasure is present. The

k

s

-state agents should either find the treasure, or state There is no treasure within distance

d

from the origin. In either case, all agents must terminate. The goal is to maximize

d

as a function of

k

and

s

.

在本文中，我们开始了基于agent的忙狸问题的研究：考虑一个由无限离散线和一组k个不同的（每个都有自己的算法）agent组成的同步系统，每个agent都是一个s状态自动机，只能看到共定位agent的状态（和id）。假设所有代理开始时都位于同一位置，那么这些代理可以执行的最长终止计算的长度是多少？我们给出了k=2时的渐近最优答案Θ（s3），证明了k≥3时答案不是一个可计算的函数，给出了k=3时至少达到22Ωs0.347步长的确定性构造，并证明了3个智能体可以模拟一个图灵机。我们还表明，任何s状态图灵机都可以由O（log）个无关代理组成的团队模拟。作为我们结果的一个应用，我们展示了如何解决寻宝问题的以下变体：不能保证宝藏存在。k个状态代理要么找到宝藏，要么状态为离原点距离d内没有宝藏。在任何一种情况下，所有代理都必须终止。我们的目标是最大化d作为k和s的函数。

{"title":"Busy agents on a line","authors":"Stefan Dobrev , Rastislav Královič , Dana Pardubská","doi":"10.1016/j.dam.2025.11.049","DOIUrl":"10.1016/j.dam.2025.11.049","url":null,"abstract":"<div><div>In this paper we initiate the study of agent-based busy beaver problem: Consider a synchronous system consisting of an infinite discrete line and a group of <math><mi>k</mi></math> distinct (each with its own algorithm) agents, each being an <math><mi>s</mi></math>-state automaton that can see only the states (and IDs) of the co-located agents. Assuming that all agents start co-located, what is length of the longest terminating computation these agents can perform?</div><div>We give asymptotically optimal answer of <math><mrow><mi>Θ</mi><mrow><mo>(</mo><msup><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math>, show that the answer is not a computable function for <math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math>, give a deterministic construction for <math><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></math> achieving at least <math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>Ω</mi><mfenced><mrow><msup><mrow><mi>s</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>347</mn></mrow></msup></mrow></mfenced></mrow></msup></mrow></msup></math> steps, and show that 3 agents can simulate a Turing machine. We also show that any <math><mi>s</mi></math>-state Turing machine can be simulated by a team of <math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mi>s</mi><mo>)</mo></mrow></mrow></math> oblivious agents.</div><div>As an application of our results, we show how to solve the following variant of the Treasure Hunt problem: There is no guarantee that the treasure is present. The <math><mi>k</mi></math>\u0000 <math><mi>s</mi></math>-state agents should either find the treasure, or state There is no treasure within distance <math><mi>d</mi></math> from the origin. In either case, all agents must terminate. The goal is to maximize <math><mi>d</mi></math> as a function of <math><mi>k</mi></math> and <math><mi>s</mi></math>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 214-233"},"PeriodicalIF":1.0,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685320","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}

引用次数: 0

ILP and CP models for normalized integer weighted voting game design problem 正则化整数加权投票博弈设计问题的ILP和CP模型

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-04 DOI: 10.1016/j.dam.2025.11.051

Roger Kameugne , Pierre Wambo , Yves Pascal Ndjopnang Wantiep

Weighted voting games are a family of cooperative games, where each voter or player carries a weight. If the sum of the weights of the players in favor of a proposal is larger than or equal to a given quota, then this proposal is accepted. In the weighted game field, the problem of normalized integer weighted voting game representation (NIWVG), and the integer weighted voting game design problem (IWVGD) or inverse power index problem have been subjects of several studies. A solution to the last problem is an integer-weighted voting game with a specific power index, as close as possible (following a norm) to a given influence vector. Finding a normalized integer-weighted voting game solution for the inverse power index problem is a challenging task. The nature of the power index and the norm awarded in the problem can impact the approach used to solve this problem. In this paper, exact approaches based on integer linear programming (ILP) and constraint programming (CP) are proposed for this problem. The models proposed in this paper combine the Shapley–Shubik index or Banzhaf–Coleman index with Manhattan and Chebyshev norms respectively. Computational results show that ILP is an adequate approach when the power index and the norm can be linearized while CP appears to be suitable for the remaining cases of voting games with a small number of players.

加权投票游戏是一种合作游戏，其中每个投票人或玩家都有自己的权重。如果赞成某一提议的玩家的权重之和大于或等于给定的配额，则该提议被接受。在加权博弈领域，正则化整数加权投票博弈表示问题（NIWVG）和整数加权投票博弈设计问题（IWVGD）或幂指数逆问题已经成为许多研究的主题。最后一个问题的解决方案是使用具有特定权力指数的整数加权投票游戏，尽可能接近给定的影响向量（遵循规范）。幂指数反比问题的归一化整数加权投票博弈解是一项具有挑战性的任务。问题中功率指标的性质和授予的规范会影响解决该问题的方法。本文提出了基于整数线性规划（ILP）和约束规划（CP）的精确求解方法。本文提出的模型分别将Shapley-Shubik指数或Banzhaf-Coleman指数与Manhattan和Chebyshev范数相结合。计算结果表明，当幂指数和范数可以线性化时，ILP是一种合适的方法，而CP则适用于参与者较少的投票博弈的剩余情况。

{"title":"ILP and CP models for normalized integer weighted voting game design problem","authors":"Roger Kameugne , Pierre Wambo , Yves Pascal Ndjopnang Wantiep","doi":"10.1016/j.dam.2025.11.051","DOIUrl":"10.1016/j.dam.2025.11.051","url":null,"abstract":"<div><div>Weighted voting games are a family of cooperative games, where each voter or player carries a weight. If the sum of the weights of the players in favor of a proposal is larger than or equal to a given quota, then this proposal is accepted. In the weighted game field, the problem of normalized integer weighted voting game representation (NIWVG), and the integer weighted voting game design problem (IWVGD) or inverse power index problem have been subjects of several studies. A solution to the last problem is an integer-weighted voting game with a specific power index, as close as possible (following a norm) to a given influence vector. Finding a normalized integer-weighted voting game solution for the inverse power index problem is a challenging task. The nature of the power index and the norm awarded in the problem can impact the approach used to solve this problem. In this paper, exact approaches based on integer linear programming (ILP) and constraint programming (CP) are proposed for this problem. The models proposed in this paper combine the Shapley–Shubik index or Banzhaf–Coleman index with Manhattan and Chebyshev norms respectively. Computational results show that ILP is an adequate approach when the power index and the norm can be linearized while CP appears to be suitable for the remaining cases of voting games with a small number of players.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 197-209"},"PeriodicalIF":1.0,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685318","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}

引用次数: 0

HV-symmetric polyhedra and bipolarity 高压对称多面体和双极性体

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-04 DOI: 10.1016/j.dam.2025.10.062

David Avis

A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron

P

in

R^{n}

can be described by an

H - r e p r e s e n t a t i o n

H (P)

consisting of half spaces or equivalently by a

V

-representation

V (P)

consisting of the convex hull of a set of vertices and extreme rays. We can define matrices

H (P)

and

V (P)

, each with

n + 1

columns, that encode these representations. Define polyhedron

Q

by setting

H (Q) = V (P)

. We show that

Q

is the polar of

P

. Call

P

H V

-symmetric if

Q

is pointed and

V (Q)

in turn encodes

H (P)

. It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are

H V

-symmetric. We show here that, more generally, a pointed polyhedron with pointed polar is

H V

-symmetric if and only if it contains the origin. We prove this using Minkowski’s bipolar equation and discuss implications for the vertex and facet enumeration problems.

如果多面体至少包含一个顶点，则多面体是有点的。Rn中的每一个点多面体P都可以用由半空间组成的H -表示H(P)来描述，或者等价地用由一组顶点和极值射线组成的凸包组成的V-表示V(P)来描述。我们可以定义矩阵H(P)和V(P)，每个都有n+1列，来编码这些表示。设H(Q)=V(P)定义多面体Q。我们证明了Q是P的极，如果Q是尖的，并且V(Q)反过来编码H(P)，则称P为hv对称的。众所周知，在其内部包含原点和尖多面体锥的多面体是hv对称的。我们在这里证明，更一般地说，一个尖极的尖多面体当且仅当它包含原点时是hv对称的。我们用Minkowski的双极方程证明了这一点，并讨论了顶点和面枚举问题的意义。

{"title":"HV-symmetric polyhedra and bipolarity","authors":"David Avis","doi":"10.1016/j.dam.2025.10.062","DOIUrl":"10.1016/j.dam.2025.10.062","url":null,"abstract":"<div><div>A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron <math><mi>P</mi></math> in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> can be described by an <math><mrow><mi>H</mi><mo>−</mo><mi>r</mi><mi>e</mi><mi>p</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></math>\u0000 <math><mrow><mi>H</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math> consisting of half spaces or equivalently by a <math><mi>V</mi></math>-representation <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math> consisting of the convex hull of a set of vertices and extreme rays. We can define matrices <math><mrow><mi>H</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math>, each with <math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math> columns, that encode these representations. Define polyhedron <math><mi>Q</mi></math> by setting <math><mrow><mi>H</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math>. We show that <math><mi>Q</mi></math> is the polar of <math><mi>P</mi></math>. Call <math><mi>P</mi></math>\u0000 <math><mrow><mi>H</mi><mi>V</mi></mrow></math>-symmetric if <math><mi>Q</mi></math> is pointed and <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math> in turn encodes <math><mrow><mi>H</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math>. It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are <math><mrow><mi>H</mi><mi>V</mi></mrow></math>-symmetric. We show here that, more generally, a pointed polyhedron with pointed polar is <math><mrow><mi>H</mi><mi>V</mi></mrow></math>-symmetric if and only if it contains the origin. We prove this using Minkowski’s bipolar equation and discuss implications for the vertex and facet enumeration problems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 210-213"},"PeriodicalIF":1.0,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685322","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}

引用次数: 0

IP.LSH.DBSCAN: Integrated parallel density-based clustering by locality-sensitive hashing IP.LSH.DBSCAN：通过位置敏感散列集成并行基于密度的集群

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-01 DOI: 10.1016/j.dam.2025.11.047

Amir Keramatian, Vincenzo Gulisano, Marina Papatriantafilou, Philippas Tsigas

Locality-sensitive hashing (LSH) is an established method for fast data indexing and approximate similarity search, with useful parallelism properties. Although indexes and similarity measures are key for data clustering, little has been investigated on the multifaceted benefits of LSH in the problem. We show how approximate DBSCAN clustering can be fused into the process of creating an LSH index, and, through parallelization and fine-grained synchronization, also utilize efficiently available computing capacity. The resulting algorithm,

I P . L S H . D B S C A N

, described in this article, can support a wide range of applications with diverse distance functions, as well as data distributions and dimensionality. We analyse the algorithm’s asymptotic completion time and provide an open-source prototype implementation. We also conduct a detailed evaluation measuring latency and accuracy metrics of

I P . L S H . D B S C A N

, on a 36-core machine with 2-way hyper threading on massive data-sets with various numbers of dimensions. The analysis and the empirical study of

I P . L S H . D B S C A N

show how it complements the landscape of established state-of-the-art methods, by offering up to several orders of magnitude speed-up on higher dimensional datasets, with tunable high clustering accuracy.

位置敏感散列（LSH）是一种成熟的快速数据索引和近似相似性搜索方法，具有有用的并行性。虽然索引和相似性度量是数据聚类的关键，但很少有人研究LSH在问题中的多方面好处。我们将展示如何将近似DBSCAN聚类融合到创建LSH索引的过程中，并通过并行化和细粒度同步，有效地利用可用的计算能力。得到的算法IP.LSH。本文中描述的DBSCAN可以支持具有不同距离函数以及数据分布和维度的广泛应用程序。我们分析了算法的渐近完成时间，并提供了一个开源原型实现。我们还进行了详细的评估，测量IP.LSH的延迟和准确性指标。DBSCAN，在36核机器上使用双向超线程处理具有不同维数的大量数据集。IP.LSH.DBSCAN的分析和实证研究表明，通过在高维数据集上提供高达几个数量级的加速，并具有可调的高聚类精度，它如何补充了已建立的最先进的方法。

{"title":"IP.LSH.DBSCAN: Integrated parallel density-based clustering by locality-sensitive hashing","authors":"Amir Keramatian, Vincenzo Gulisano, Marina Papatriantafilou, Philippas Tsigas","doi":"10.1016/j.dam.2025.11.047","DOIUrl":"10.1016/j.dam.2025.11.047","url":null,"abstract":"<div><div>Locality-sensitive hashing (LSH) is an established method for fast data indexing and approximate similarity search, with useful parallelism properties. Although indexes and similarity measures are key for data clustering, little has been investigated on the multifaceted benefits of LSH in the problem. We show how approximate DBSCAN clustering can be fused into the process of creating an LSH index, and, through parallelization and fine-grained synchronization, also utilize efficiently available computing capacity. The resulting algorithm, <math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math>, described in this article, can support a wide range of applications with diverse distance functions, as well as data distributions and dimensionality. We analyse the algorithm’s asymptotic completion time and provide an open-source prototype implementation. We also conduct a detailed evaluation measuring latency and accuracy metrics of <math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math>, on a 36-core machine with 2-way hyper threading on massive data-sets with various numbers of dimensions. The analysis and the empirical study of <math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math> show how it complements the landscape of established state-of-the-art methods, by offering up to several orders of magnitude speed-up on higher dimensional datasets, with tunable high clustering accuracy.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 183-196"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685323","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}

引用次数: 0

Cycles and trees in randomly perturbed sparse digraphs 随机扰动稀疏有向图中的圈和树

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-01 DOI: 10.1016/j.dam.2025.11.046

Yulin Chang , Jichang Wu , Zhiwei Zhang

We study the existence of directed Hamilton cycles and oriented spanning trees of bounded degree in randomly perturbed sparse digraphs. Let

D_{α}

be a digraph on

n

vertices with minimum semi-degree

δ^{0} (D_{α}) \geq α n

, where

0 < α < 1

, and let

D (n, p)

be the binomial random digraph on

n

vertices in which each of

n (n - 1)

possible ordered pairs forms an arc independently with probability

p

. We prove that, for any

α = α (n) : N \mapsto (0, 1)

, if

β = (6 + o (1)) log \frac{1}{α}

, then

D_{α} \cup D (n, β / n)

contains a directed Hamilton cycle with high probability. Moreover, if

α = ω (n^{- 1 / 8})

, and

β = β (n) = 6 log \frac{1}{α}

, then

D_{α} \cup D (n, β / n)

is vertex-pancyclic with high probability. Finally, we also prove a similar result for oriented spanning trees in this model.

研究了随机摄动稀疏有向图中有向Hamilton环和有界度有向生成树的存在性。设Dα是n个顶点上最小半度δ0(Dα)≥αn的有向图，其中0<；α<1；设D（n,p）是n个顶点上的二项随机有向图，其中n（n−1）个可能的有序对中的每一个都以概率p独立形成一个弧。我们证明了对于任意α=α(n): n =(0,1)，若β=(6+o(1))log1α，则Dα∪D（n,β/n）包含一个高概率的有向Hamilton环。如果α=ω(n−1/8),β=β(n)=6log1α，则Dα∪D（n,β/n）是高概率顶点泛环。最后，我们也证明了该模型中有向生成树的类似结果。

引用次数: 0

The lower bounds of 4-tree connectivity of Cartesian product graphs 笛卡尔积图的四树连通性下界

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-29 DOI: 10.1016/j.dam.2025.11.050

Lin Li , Rong-Xia Hao , Yan-Quan Feng , Jaeun Lee , Eddie Cheng

<div><div>Let <math><mi>G</mi></math> be a graph with vertex set <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> and edge set <math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. Two trees <math><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and <math><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub></math> of <math><mi>G</mi></math> are internally disjoint <math><mi>S</mi></math>-tree for <math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> if <math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mo>0̸</mo></mrow></math> and <math><mrow><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>∩</mo><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>S</mi></mrow></math>. The <math><mi>r</mi></math>-tree connectivity <math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> of <math><mi>G</mi></math> is <math><mrow><mo>min</mo><mrow><mo>{</mo><msub><mrow><mi>κ</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>r</mi><mo>}</mo></mrow></mrow></math>, where <math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> denotes the maximum number of internally pairwise disjoint <math><mi>S</mi></math>-trees in <math><mi>G</mi></math>. The lower bound of <math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> for Cartesian product graph <math><mrow><mi>G</mi><mo>□</mo><mi>H</mi></mrow></math> of connected graphs <math><mi>G</mi></math> and <math><mi>H</mi></math> is characterized in Gao et al. (2018). In this paper, we obtained a lower bound of <math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>4</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> which depends only on the <math><mi>k</mi></math>-tree connectivity of <math><mi>G</mi></math> and <math><mi>H</mi></math> with <math><mrow><mi>k</mi><mo>≤</mo><mn>4</mn></mrow></math>. Fur

设G是一个顶点集V(G)，边集E(G)的图。如果E(Ti)∩E(Tj)=0，且V(Ti)∩V(Tj)=S，则G的两棵树Ti和Tj是S≤V(G)的S树。G的r树连通性κr(G)为min{κS(G)|S≤V(G),|S|=r}，其中κS(G)表示G中最大内部两两不相交S树数。高等（2018）描述了连通图G和H的笛卡尔积图G□H的κ3（G□H）下界。在本文中，我们得到了一个κ4（G□H）的下界，它只依赖于G和H在k≤4时的k树连通性。进一步，对该界的紧性进行了分析。

引用次数: 0

Two extremal problems for 4-cycles in 4-partite graphs 四部图中4环的两个极值问题

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-28 DOI: 10.1016/j.dam.2025.11.048

Zemin Jin, Huifang Liu, Qing Jie

<div><div>In this paper, we consider two extremal problems about 4-cycles in multipartite graphs. Denote by <math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></msubsup></math> the 4-cycle in a multipartite graph whose vertices come from exactly four different partite sets. We call a 4-cycle in a multipartite graph multipartite, denoted by <math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>m</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi></mrow></msubsup></math>, if its vertices come from at least three different partite sets. An edge-colored graph is called rainbow if any two edges of it have different colors. For given graphs <math><mi>G</mi></math> and <math><mi>H</mi></math>, the anti-Ramsey number <math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> is the maximum number of colors in an edge-colored <math><mi>G</mi></math> with no rainbow <math><mi>H</mi></math>. This graph parameter is closely related to the Turán number. These two parameters on 3-cycles in general complete multipartite graphs have been well determined. The anti-Ramsey number on 4-cycles was solved in general complete <math><mi>r</mi></math>-partite graphs, while the number on multipartite 4-cycle was only determined for <math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math>. The Turán number on (multipartite) 4-cycles in complete <math><mi>r</mi></math>-partite graphs was proved only for <math><mrow><mi>r</mi><mo>≤</mo><mn>3</mn></mrow></math>. In this paper, we show that <math><mrow><mi>e</mi><mi>x</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> and <math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><m

本文研究了多部图中关于4环的两个极值问题。用C4(4)表示顶点恰好来自四个不同部集的多部图的4环。我们称多部图中的一个4环为多部图，用C4multi表示，如果它的顶点至少来自三个不同的部集。如果任意两条边的颜色不同，则称为彩虹。对于给定的图G和图H，反拉姆齐数AR（G，H）是没有彩虹H的边色图G的最大颜色数。该图参数与Turán数密切相关。这两个参数在一般完全多部图的3环上已经得到了很好的确定。4环上的反拉姆齐数在一般完全r部图中都能求出，而多部4环上的反拉姆齐数只有在r=3时才能确定。完全r部图中（多部）4环的Turán个数仅在r≤3时得到证明。在本文中，我们证明了ex(Kn1,n2,n3,n4,C4(4))=n1n2+n1n3+n1n4+n2n3和AR(Kn1,n2,n3,n4,C4multi)=n1n2+n3n4+2，其中n1≥n2≥n3≥n4≥1。

引用次数: 0

A multivariate complexity analysis of the Generalized Noah’s Ark Problem 广义诺亚方舟问题的多元复杂性分析

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-27 DOI: 10.1016/j.dam.2025.11.037

Christian Komusiewicz , Jannik Schestag

In the Generalized Noah’s Ark Problem, one is given a phylogenetic tree on a set of species

X

and a set of conservation projects for each species. Each project comes with a cost and raises the survival probability of the corresponding species. The aim is to select a conservation project for each species such that the total cost of the selected projects does not exceed some given threshold and the expected phylogenetic diversity is as large as possible. We study the complexity of Generalized Noah’s Ark Problem and some of its special cases with respect to several parameters related to the input structure, such as the number of different costs, the number of different survival probabilities, or the number of species,

| X |

.

在广义诺亚方舟问题中，给定一组物种X的系统发育树和每个物种的保护计划。每个项目都有成本，并提高了相应物种的生存概率。其目的是为每个物种选择一个保护项目，使所选项目的总成本不超过某个给定的阈值，并使预期的系统发育多样性尽可能大。本文研究了广义诺亚方舟问题的复杂性及其一些特殊情况，这些问题涉及到与输入结构相关的几个参数，如不同成本的数量、不同生存概率的数量或物种的数量，|X|。

引用次数: 0

The σ-irregularity of trees with maximum degree 5 最大度为5的树的σ-不规则度

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-25 DOI: 10.1016/j.dam.2025.11.045

Darko Dimitrov , Žana Kovijanić Vukićević , Goran Popivoda , Jelena Sedlar , Riste Škrekovski , Saša Vujošević

The

σ

-irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph

G = (V, E)

as

σ (G) = \sum_{u v \in E} {(d (u) - d (v))}^{2}

, where

d (u)

and

d (v)

denote the degrees of vertices

u

and

v

, respectively. Recent research has successfully characterized chemical trees with the maximum

σ

-irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree

Δ

. Application of these properties enables us to characterize maximal trees with

Δ = 5 .

We establish that extremal trees contain only vertices of degrees

1,

2 and

Δ

. Moreover, the number of edges with both end-vertices having the degree 2 or

Δ

is very small, so almost all edges have the (second) maximum possible contribution to

σ

-irregularity. We believe this property or similar should extend to maximal trees for any value of

Δ

, so this is an interesting direction for further research.

σ-不规则性是已建立的Albertson不规则性的一个变体，它是对图G=（V，E）定义为σ(G)=∑uv∈E(d(u) - d(V))2的拓扑不变量，其中d(u)和d(V)分别表示顶点u和V的度数。最近的研究成功地描述了具有最大σ-不规则性的化学树。在本文中，我们通过建立具有规定最大度的极大树Δ的几个结构性质来扩展这一研究。这些性质的应用使我们能够表征Δ=5的极大树。我们建立了极值树只包含度为1、2和Δ的顶点。此外，两端顶点都为2度或Δ的边的数量非常少，所以几乎所有的边对σ-不规则度的贡献都是第二大的。我们相信这个性质或类似的性质可以推广到任何值Δ的极大树，所以这是一个有趣的进一步研究方向。

{"title":"The σ-irregularity of trees with maximum degree 5","authors":"Darko Dimitrov , Žana Kovijanić Vukićević , Goran Popivoda , Jelena Sedlar , Riste Škrekovski , Saša Vujošević","doi":"10.1016/j.dam.2025.11.045","DOIUrl":"10.1016/j.dam.2025.11.045","url":null,"abstract":"<div><div>The <math><mi>σ</mi></math>-irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph <math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math> as <math><mrow><mi>σ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, where <math><mrow><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math> denote the degrees of vertices <math><mi>u</mi></math> and <math><mi>v</mi></math>, respectively. Recent research has successfully characterized chemical trees with the maximum <math><mi>σ</mi></math>-irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree <math><mi>Δ</mi></math>. Application of these properties enables us to characterize maximal trees with <math><mrow><mi>Δ</mi><mo>=</mo><mn>5</mn><mo>.</mo></mrow></math> We establish that extremal trees contain only vertices of degrees <math><mrow><mn>1</mn><mo>,</mo></mrow></math> 2 and <math><mi>Δ</mi></math>. Moreover, the number of edges with both end-vertices having the degree 2 or <math><mi>Δ</mi></math> is very small, so almost all edges have the (second) maximum possible contribution to <math><mi>σ</mi></math>-irregularity. We believe this property or similar should extend to maximal trees for any value of <math><mi>Δ</mi></math>, so this is an interesting direction for further research.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 124-136"},"PeriodicalIF":1.0,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618503","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}

引用次数: 0