Dorota Kuziak, Ismael G. Yero
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the highest interest for the research community dealing with several variants of metric dimension in graphs.
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A combinatorial survey of identities for the double factorial
我们调查的一些双阶乘，如几十个身份的组合解释，例如，（2n-2）！+ Sum_{k=2} ^{n}（2n-1）！！（2k-4）！！/（2k-1）！！=（2n-1）！我们的方法大多是双射的。
In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, {\em Combinatorial Enumeration}, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collabor…
Computational Aspects of the Combinatorial Nullstellensatz Method
在这里，我们讨论一些计算方面的组合零stellenerance参数。我们的主要结果表明，数量级的对称群与置换的变量在代数约束，决定了算法的性能自然推导出从阿隆的组合Nullstellenerance参数。最后给出了一个证明组合问题解的存在性或不存在性的原始-对偶多项式构造。
A survey of combinatorial aspects in the topology of complex hyperplane arrangements
我们调查复杂的超平面安排的拓扑结构和组合之间的相互作用。不声称是详尽的，我们在此设置组合方面的基本群，相关的分次李代数，高同伦群，上同调环，扭曲的同源秩1复系数，和米尔诺纤维。
Overview of some general results in combinatorial enumeration
这篇综述文章致力于组合枚举的一般结果。第一部分综述了组合结构的遗传性质的增长结果。这些包括排列，有序和无序图和超图，关系结构等。第二部分概括地论述了五个问题：1.计算晶格多面体中的晶格点，2.上下文无关语言的增长，3。完整性（即，前递归性）的标号正则图的数目，4。频繁出现的渐近cn^{-3/2}r^n和5。MSOL可定义结构数量的最终模块周期性。
Combinatorial Gray codes-an updated survey
本文综述了组合Gray码研究的最新进展，更新了Savage的经典综述，涵盖多种组合对象的Gray码列表，并强调其与图论、代数等领域的联系，提出未来挑战性问题以激发新研究。
The Generalized Combinatorial Lason-Alon-Zippel-Schwartz Nullstellensatz Lemma
本文综述了Alon组合零点定理与Schwartz-Zippel引理的若干强化与推广，证明了当变量在足够大范围内独立取值时多元多项式非零值的存在性。
On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results
度量维数是一个研究得很好的图参数。最近，引入了邻接度量维数和局部度量维数。我们结合这些变种，并引入本地邻接度量维度。我们证明了一个$n$阶图和一些非平凡图$H$的冠积的（局部）度量维数等于$n$乘以$H$的（局部）邻接度量维数.这种强关系也使我们能够推断计算（本地）度量维度的计算硬度结果，根据硬度结果（本地）邻接度量维度，我们也提供。我们还研究了图的强积的组合性质，并强调不同类型的孪生子在确定特别是图的邻接度量维数中所起的作用。
Combinatorial properties of continuous graphs: A survey of challenges, solutions and open problems
本文聚焦连续图（边为单位长度连续区间）上的组合优化问题，综述了独立集、顶点覆盖等问题的已知研究成果，并整理了相关开放问题清单，为该领域研究提供参考。
Many traditional algorithms for solving combinatorial optimization problems involve using hand-crafted heuristics that sequentially construct a solution. Such heuristics are designed by domain experts and may often be suboptimal due to the hard nature of the problems. Reinforcement learning (RL) proposes a good alternative to automate the search of these heuristics by training an agent in a supervised or self-supervised manner. In this survey, we explore the recent advancements of applying RL frameworks to hard combinatorial problems. Our survey provides the necessary background for operations…
Combinatorial Designs and Cellular Automata: A Survey
本文综述将元胞自动机视为代数系统，梳理利用双置换 CA 构造相互正交拉丁方的线性与非线性结果及其密码学应用，并指出未来开放问题。
A subset $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total dominating set $D$ is called a total co-independent dominating set if the subgraph induced by $V-D$ is edgeless and has at least one vertex. The minimum cardinality of any total co-independent dominating set is the total co-independent domination number of $G$ and is denoted by $γ_{t,coi}(G)$. In this work we study some complexity and combinatorial properties of $γ_{t,coi}(G)$. Specifically, we prove that deciding whether $γ_{t,coi}(G)\le k$ for a given …
Triangulated Manifolds with Few Vertices: Combinatorial Manifolds
在三角流形的组合性质综述中，我们讨论了单纯流形和组合流形的顶点数的各种下界。此外，我们给出了所有已知的顶点最小三角剖分的例子。
Grothendieck-type inequalities in combinatorial optimization
我们调查的Grothendieck不等式和它的变种，组合优化和计算复杂性的连接。
Heffter arrays were introduced by Archdeacon in 2015 as an interesting link between combinatorial designs and topological graph theory. Since the initial paper on this topic, there has been a good deal of interest in Heffter arrays as well as in related topics such as the sequencing of subsets of a group, biembeddings of cycle systems on a surface, and orthogonal cycle systems. This survey presents an overview of the current state of the art of this topic. We begin with an introduction to Heffter arrays for the reader who is unfamiliar with the subject, then we give a unified and comprehensive…
Groups and Combinatorial Number Theory
在这次演讲中，我们介绍了几个主题在组合数论这是相关的群体，这些主题包括组合方面的覆盖群的陪集，也限制和集和零和问题的阿贝尔群。一个调查的已知结果和公开问题的主题是在一个流行的方式。
Combinatorial Representation Theory
我们试图调查领域的组合表示论，描述的主要结果和主要问题，并给出了更新的现状。我们在这个领域给出了一个个人观点，同时意识到有很多重要而美丽的工作，我们还没有能够提到。
This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is entirely expository (and written to a large extent as a repository for folklore proofs); no new results (and few, if any, new proofs) appear.
In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-Émery curvature, and Ollivier's Ricci curvature. For each curvature notion, the definition and its motivation from Riemannian geometry will be explained. Moreover, we bring together some global results and geometric concepts in Riemannian geometry that are related to curvature (e.g. Bonnet-Myers theorem, Laplacian operator, Lichnerowicz theorem, Cheeger constant), and then compare them to the discrete analogues in some (if not all) of the discrete curvature notions. …
The Neighbor-Joining algorithm is a popular distance-based phylogenetic method that computes a tree metric from a dissimilarity map arising from biological data. Realizing dissimilarity maps as points in Euclidean space, the algorithm partitions the input space into polyhedral regions indexed by the combinatorial type of the trees returned. A full combinatorial description of these regions has not been found yet; different sequences of Neighbor-Joining agglomeration events can produce the same combinatorial tree, therefore associating multiple geometric regions to the same algorithmic output. …
In this survey we discuss the notion of combinatorial interpretation in the context of Algebraic Combinatorics and related areas. We approach the subject from the Computational Complexity perspective. We review many examples, state a workable definition, discuss many open problems, and present recent results on the subject.
A combinatorial introduction to Adinkras
本文综述了用于解决超对称微分方程的图形工具Adinkra的组合学，将其表征为特殊的1-因子分解，并首次利用拉丁矩形对其进行刻画。
A Combinatorial Problem Related to Sparse Systems of Equations
稀疏方程组在科学和工程中经常出现。在这方面的贡献，我们处理稀疏系统中常见的密码分析。给定一个密码系统，将其转换为稀疏方程组，然后求解该系统以检索密钥或明文。Raddum和Semaev提出了解决这种稀疏系统的新方法。事实证明，组合MaxMinMax问题提供了稀疏系统的平均计算复杂性的界限。在本文中，我们开始研究这个MaxMinMax问题的线性代数变化。
Combinatorics of slices of cubes
本文通过结合组合、代数与数值方法，对维度至六的立方体超平面切片进行了完整计算分类，确定了各维度类型数，并扩展至七维过原点情形，同时分析了分布并提出了新理论结果与猜想。
Pseudo-Triangulations - a Survey
伪三角形是具有三个凸顶点的简单多边形，而伪三角剖分是将平面区域面对面平铺成伪三角形。伪三角剖分在计算几何中作为数据结构出现，在刚性理论中作为平面杆-关节框架出现，在局部凸曲面上作为投影出现。这一调查目前的文献包括组合性质和计数的特殊类，刚性理论的结果，表示为多面体，直线图纸从抽象版本称为组合伪三角，算法和应用程序的伪三角。
Remarks on Combinatorial Aspects of the KP Equation
我们调查了几个结果连接组合学和朗斯基解的KP方程，上下文的成功最近的方法介绍了儿玉等。我们包括必要的组合和分析背景，提出了一个公式的广义KP孤子，计算几个明确的例子，并指出如何这样的角度可以用来扩展以前的研究相关的线孤子解的KP方程与格拉斯曼。
Pseudorandomness and Combinatorial Constructions
在组合数学中，概率方法是一个非常强大的工具来证明组合对象的存在性，具有有趣和有用的性质。具有这种性质的对象的显式构造通常非常困难，或者是未知的。在计算机科学中，对于同一问题，概率算法有时比已知的确定性算法更简单和更有效。尽管有随机选择的力量的证据，伪随机性的计算理论表明，在某些复杂性理论的假设下，每一个概率算法都有一个有效的确定性模拟和大量的概率方法的应用可以转换为显式的结构。在这篇综述论文中，我们描述了伪随机性计算理论的条件"去随机化“结果和某些组合对象（如纠错码和"随机性提取器”）的无条件显式构造之间的联系。''
Perfect graphs: a survey
完美图是由Claude Berge在20世纪60年代定义的。它们是图论、线性规划和组合优化的重要研究对象。Claude Berge提出了一个猜想，这个猜想在2002年被Chudnovsky，Robertson，Seymour和Thomas证明，现在被称为强完美图定理。这是一个关于完美图的综述，主要集中在强完美图定理上。
We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also…
Some Algorithmic Problems in Polytope Theory
本文综述了关于凸多胞体的组合性质和几何性质的算法问题。我们列出了35个问题，每个问题的理论复杂性状态的当前状态的知识报告。这些问题被分为“坐标描述”、“组合结构”、“同构”、“优化”、“可实现性”和“超越多面体”等部分。
Lecture hall partitions are a fundamental combinatorial structure which have been studied extensively over the past two decades. These objects have produced new results, as well as reinterpretations and generalizations of classicial results, which are of interest in combinatorial number theory, enumerative combinatorics, and convex geometry. In a recent survey of Savage \cite{Savage-LHP-Survey}, a wide variety of these results are nicely presented. However, since the publication of this survey, there have been many new developments related to the polyhedral geometry and Ehrhart theory arising …
We survey some general-purpose symbolic software packages that implement algorithms from enumerative and analytic combinatorics. Software for the following areas is covered: basic combinatorial objects, symbolic combinatorics, Pólya theory, combinatorial species, and asymptotics. We describe the capabilities that the packages offer as well as some of the algorithms used, and provide links to original documentation. Most of the packages are freely downloadable from the web.
Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Qua…
Free Calculus
我给一个调查我的工作组合和概率方面的自由概率论。特别是，我以自由累积量的形式给出了自由度的组合描述，并给出了我与Philippe Biane联合研究自由布朗运动随机积分理论的主要结果的一些想法。
Honey from the Hives: A Theoretical and Computational Exploration of Combinatorial Hives
本文探讨了通过优化方案从Hermitian矩阵对生成蜂巢的理论与计算方法，提出新的算法并验证了两种数值算法在估计Littlewood-Richardson系数中的有效性。
It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is a family of plabic graphs giving a cluster structure of the same Grassmannian. Such plabic graphs are related by the operation called square move which can be considered as the mutation in cluster theory. By using a plabic graph, we also obtain the Newton--Okounkov polytope which gives a toric degeneration of the Grassmannian. The purposes of this article i…
Computational Complexity in Algebraic Combinatorics
本文探讨代数组合学优美乘积公式（如Kronecker和plethysm系数）的计算复杂性，并指出该研究推翻了Geometric Complexity Theory解决VP vs VNP问题的设想。
Lecture notes on quantum cohomology of the flag manifold
这是一个博览会的一些最近的发展有关的对象的标题，特别是组合计算的（亏格0）Gromov-Witten不变量的旗流形和二次代数方法。这些笔记主要是根据我与S.Gelfand、A.N.Kirillov和A. Postnikov的联合论文。这绝不是一个详尽的调查的主题，而是一个偶然的介绍其组合方面。
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