Shmuel Onn
We consider a problem of optimizing convex functionals over matroid bases. It is richly expressive and captures certain quadratic assignment and clustering problems. While generally NP-hard, we show it is polynomial time solvable when a suitable parameter is restricted.
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Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M${}^\natural$-convex function as discrete convex functions, which are generalizations of valuated matroids due to Dress and Wenzel (1992). In the present paper we consider a new operation defined by a convolution of sections of an M${}^\natural$-convex function that transforms the given M${}^\natural$-convex function to an M-convex function, which we call a compression of an M${}^\natural$-convex function. For the class of valuated generalized matroids, which are special M${}^\natural$-convex functions, t…
In 1980, Las Vergnas defined a notion of discrete convexity for oriented matroids, which Edelman subsequently related to the theory of anti-exchange closure functions and convex geometries. In this paper, we use generalized matroid activity to construct a convex geometry associated with an ordered, unoriented matroid. The construction in particular yields a new type of representability for an ordered matroid defined by the affine representability of its corresponding convex geometry. The lattice of convex sets of this convex geometry induces an ordering on the matroid independent sets which ex…
For two matroids $M_1$ and $M_2$ with the same ground set $V$ and two cost functions $w_1$ and $w_2$ on $2^V$, we consider the problem of finding bases $X_1$ of $M_1$ and $X_2$ of $M_2$ minimizing $w_1(X_1)+w_2(X_2)$ subject to a certain cardinality constraint on their intersection $X_1 \cap X_2$. For this problem, Lendl, Peis, and Timmermans (2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $|X_1 \cap X_2|\le k$ or $|X_1 \cap X_2|\ge k$; and designed a new primal-dual algorithm for the case where…
The principle of inclusion-exclusion is applied to subsets of maximal covectors contained in halfspaces of a simple oriented matroid and to convex subsets of its ground set for enumerating tope committees.
Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive "generalized Gao-trees". We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the $s-t$ path TSP.
A note on embracing exchange sequences in oriented matroids
本文探讨了在定向模体中基之间交换序列的最小长度问题，并证明了在定向图形模体中的情况，同时研究了该问题与无向模体中长期存在的开放性问题的联系。
We show that finding lexicographically minimal $n$ bases in a matroid can be done in polynomial time in the oracle model. This follows from a more general result that the shifted problem over a matroid can be solved in polynomial time as well.
We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.
Oriented Matroids and Combinatorial Neural Codes
本文将偏序体理论与组合神经码理论联系起来，证明了判断组合码是否凸是NP难问题。
A convex ear decomposition of the augmented Bergman complex of a matroid
本文证明任意拟阵的增广Bergman复形具有凸耳分解，进而推出其双重Cohen–Macaulay性与顶部重h-向量，并给出h-多项式计算公式及非对数凹、非单峰的反例。
Bi-objective optimization problems on matroids are in general intractable and their corresponding decision problems are in general NP-hard. However, if one of the objective functions is restricted to binary cost coefficients the problem becomes efficiently solvable by an exhaustive swap algorithm. Binary cost coefficients often represent two categories and are thus a special case of ordinal coefficients that are in general non-additive. In this paper we consider ordinal objective functions with more than two categories in the context of matroid optimization. We introduce several problem varian…
Quotients of M-convex sets and M-convex functions
该研究在离散凸分析框架下统一研究拟阵等对象的商与子模函数强映射，给出M-凸集商的十种等价刻画，首次研究M-凸函数的商并构建四类刻画层级，深化了对归纳运算及连接集、函数结构的理解。
Like most multiobjective combinatorial optimization problems, biobjective optimization problems on matroids are in general intractable and their corresponding decision problems are in general NP-hard. In this paper, we consider biobjective optimization problems on matroids where one of the objective functions is restricted to binary cost coefficients. We show that in this case the problem has a connected efficient set with respect to a natural definition of a neighborhood structure and hence, can be solved efficiently using a neighborhood search approach. This is, to the best of our knowledge,…
Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning. The current best results include an $e^{2k}$-estimation for any matroid of rank $k$ and a $(1+ε)^d$-approximation for a uniform matroid of rank $k\ge d+\frac dε$, where the rank $k\ge d$ denotes the desired size of the optimal set. Our main result is a new approximation algorit…
On the Reduciblity of a Certain Type of Rank 3 Uniform Oriented Matroid by a Point
本文研究了凸n边形内部区域与对称群S_n中特定类型的n循环的关系，发现并非所有区域标签的循环都出现在每个凸n边形中，进而定义了确定性和不定性循环，并探讨了其在均匀秩3凸定向矩阵体的一点扩展中的可约性。
The multiple exchange property for matroid bases is generalized for valuated matroids and M$^\natural$-concave set functions. The proof is based on the Fenchel-type duality theorem in discrete convex analysis. The present result has an implication in economics: The strong no complementarities (SNC) condition of Gul and Stacchetti is in fact equivalent to the gross substitutes (GS) condition of Kelso and Crawford.
Rigid Matroid Categories
本研究提出拟阵的三种合成形式，将双拟阵范畴拓展为刚性幺半范畴，证明删除收缩、Tutte多项式等多种经典拟阵构造在该范畴中具有函子性或可由态射定义，为拟阵理论提供了统一的范畴论视角。
Matroid is a generalization of many fundamental objects in combinatorial mathematics , and matroid intersection problem is a classical subject in combinatorial optimization . However , only the intersection of two matroids are well understood . The solution of the intersection problem of more than three matroids is proved to be \textbf{NP-hard} . We will give a lower bound estimate on the maximal cardinality of the common independent sets in matroid intersections . We will also study some properties of the intersection of more than two matroids and deduce some analogous results for Edmonds' Mi…
We address optimization of nonlinear functions of the form $f(Wx)$, where $f:\R^d\to \R$ is a nonlinear function, $W$ is a $d\times n$ matrix, and feasible $x$ are in some large finite set $F$ of integer points in $\R^n$. One motivation is multi-objective discrete optimization, where $f$ trades off the linear functions given by the rows of $W$. Another motivation is to extend known results about polynomial-time linear optimization over discrete structures to nonlinear optimization. We assume that the convex hull of $F$ is well-described by linear inequalities. For example, the set of character…
Optimization of the directed spanning trees using the weighted matroid intersection algorithm
研究给定样本树权重等变化时更新有向最小生成树的问题，核心方法是用加权拟阵交算法，通过维护动态辅助图实现迭代更新，贡献是给出算法实现及实验分析。
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…
On Optimization and Counting of Non-Broken Bases of Matroids
该研究针对一般拟阵的NBC基开展计数与优化研究，证明最大权重NBC基求解、特定约束下NBC基计数均为NP难问题，揭示其与独立集复形的本质差异，并佐证相关上下游走可能无法快速混合。
The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems ov…
Determinant maximization provides an elegant generalization of problems in many areas, including convex geometry, statistics, machine learning, fair allocation of goods, and network design. In an instance of the determinant maximization problem, we are given a collection of vectors $v_1,\ldots, v_n \in \mathbb{R}^d$, and the goal is to pick a subset $S\subseteq [n]$ of given vectors to maximize the determinant of the matrix $\sum_{i \in S} v_iv_i^\top$, where the picked set of vectors $S$ must satisfy some combinatorial constraint such as cardinality constraint ($|S| \leq k$) or matroid constr…
A great variety of fundamental optimization and counting problems arising in computer science, mathematics and physics can be reduced to one of the following computational tasks involving polynomials and set systems: given an $m$-variate real polynomial $g$ and a family of subsets $B$ of $[m]$, (1) find $S\in B$ such that the monomial in $g$ corresponding to $S$ has the largest coefficient in $g$, or (2) compute the sum of coefficients of monomials in $g$ corresponding to all the sets in $B$. Special cases of these problems, such as computing permanents, sampling from DPPs and maximizing subde…
Signed tropical halfspaces and convexity
本文将热带凸性理论扩展至热带正象限之外，研究了符号热带数的TO-convexity与TC-convexity概念，推导了分离结果，并揭示了TC-convexity与oriented matroids表示及real tropical hyperfield线性空间的自然联系。
Given a dissimilarity map $δ$ on finite set $X$, the set of ultrametrics (equidistant tree metrics) which are $l^\infty$-nearest to $δ$ is a tropical polytope. We give an internal description of this tropical polytope which we use to derive a polynomial-time checkable test for the condition that all ultrametrics $l^\infty$-nearest to $δ$ have the same tree structure. It was shown by Ardila and Klivans \cite{ardila-klivans2006} that the set of all ultrametrics on a finite set of size $n$ is the Bergman fan associated to the matroid underlying the complete graph on $n$ vertices. Therefore, we de…
This thesis is basically devoted to matroids -- fundamental structure of combinatorial optimization -- though some of our results concern simplicial complexes, or Euclidean spaces. We study old and new problems for these structures, with combinatorial, algebraic, and topological flavor.
Reconfiguration of basis pairs in regular matroids
本文从优化角度验证了正则拟阵中长度为二的基序列的White猜想，并给出了多项式时间算法，首次为基对的交换距离提供多项式上界，同时验证了Gabow猜想。
We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.
Cone and constrained colorful Carathéodory Theorems
本文在拓扑学上扩展霍尔姆森等人的方法，研究凸锥取代凸包及横截约束非由拟阵编码时的彩色卡拉西奥多里型结果，通过分析单纯复形同调群，如定向拟阵支撑复形，实现经典特弗伯格定理新扩展及方法精细应用。
The basis exchange axiom has been a driving force in the development of matroid theory. However, the axiom gives only a local characterization of the relation of bases, which is a major stumbling block to further progress, and providing a global understanding of the structure of matroid bases is a fundamental goal in matroid optimization. While studying the structure of symmetric exchanges, Gabow proposed the problem that any pair of bases admits a sequence of symmetric exchanges. A different extension of the exchange axiom was proposed by White, who investigated the equivalence of compatible …
On the Congruency-Constrained Matroid Base
研究寻找标签之和满足特定同余条件的拟阵基问题，提出当模数为两素数积或素数幂时的多项式对数时间算法，并探讨其在加性组合猜想下向阿贝尔群的推广及优化版本。
An Approximation Framework for Parametric Matroid Interdiction Problems
针对多参数拟阵ℓ-阻断问题，本文提出首个可将非参数近似算法迁移至多参数场景的框架，可在非参数β-近似存在时得到近似质量任意接近β的多参数算法，并针对划分、图拟阵分别得到对应近似结果。
One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall's conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful conjecture on rearrangements of bases. In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matro…
Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond
本文研究两个拟阵中基数至少为τ的极大公共独立集枚举问题，提出多项式延迟和多项式空间算法，并扩展至拟阵匹配及三次子图小最小连通顶点覆盖枚举，同时给出带基数约束枚举转排序枚举的框架。
Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid c…
In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets $S\subset \Z^n$ of integer points presented by an oracle or by linear inequalities. Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of {\em edge complexity} of $S$, and use it to establish polynomial and constant upper bounds on the number of vertices of the projection $\conv(WS)$ and…
Optimizing Extension Techniques for Discovering Non-Algebraic Matroids
本文优化了检测非代数型体的扩展技术，并在9和10个点上发现了新的非代数型体，同时改进了秘密共享方案的信息比率下界。
Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the ra…
We give a deterministic polynomial time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank $r$ and the number of common bases of any two matroids of rank $r$. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze [ABF94] this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result: For the first, we build upon recent results of Adiprasito, Huh, and Ka…
We investigate uniqueness issues that arise in $l^\infty$-optimization to linear spaces and Bergman fans of matroids. For linear spaces, we give a polyhedral decomposition of $\mathbb{R}^n$ based on the dimension of the set of $l^\infty$-nearest neighbors. This implies that the $l^\infty$-nearest neighbor in a linear space is unique if and only if the underlying matroid is uniform. For Bergman fans of matroids, we show that the set of $l^\infty$-nearest points is a tropical polytope and give an algorithm to compute its tropical vertices. A key ingredient here is a notion of topology that gener…
Base Modulus for Matroid Truncation, Strength, and Fractional Arboricity
本文研究了Matroid截断下的基模性质，推导了通用密度概念，并通过信息论方法重新定义了该密度，同时探讨了严格同质Matroid及相关图结构优化问题。
Over the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorithms for high-dimensional polyhedra given by inequalities. It is a famous and long standing open question in polyhedral theory and computational geometry whether the vertices of a polytope (bounded polyhedron), described by a set of linear constraints, can be enumerated in total polynomial time. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra $P = \{x : A…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of ou…
Halfspace separation in geodesic convexity
研究了在弱桥图、伪模图和Matroid基图中，测地凸性半空间分离问题的多项式解法，证明其复杂度降低。
The polytope of all $q$-rank functions
本文将q秩函数对应到多面体中的点，证明该多面体不含内部格点，研究其格点凸组合性质，重点关注铺砌与均匀q-拟阵的凸组合。
Given a combinatorial optimization problem $Π$ and an increasing finite sequence $c$ of natural numbers, we obtain a cardinality constrained version $Π_c$ of $Π$ by permitting only those feasible solutions of $Π$ whose cardinalities are members of $c$. We are interested in polyhedra associated with those problems, in particular in inequalities that cut off solutions of forbidden cardinality. Maurras (1977) and Camion and Maurras (1982) introduced a family of inequalities, that we call forbidden set inequalities, which can be used to cut off those solutions. However, these inequalities are in g…
One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. Bérczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spann…
Lorentzian polynomials and matroids over triangular hyperfields 2: Analytic aspects
本文研究洛伦兹多项式与三角超域上表示的定量关系，证明对于每个matroid M，存在q>0使得P L_M位于两个薄施鲁德细胞之间，并确定q(n)的渐进行为。
Lorentzian polynomials
本文研究 Lorentzian 多项式类，证明其 Hessian 正特征值性质，刻画拟阵与 M-凸集，并应用于 Mason 猜想与 M-矩阵特征多项式。
The present work is the first member of a pair of papers concerning decreasingly-minimal (dec-min) elements of a set of integral vectors, where a vector is dec-min if its largest component is as small as possible, within this, the next largest component is as small as possible, and so on. This discrete notion, along with its fractional counterpart, showed up earlier in the literature under various names. The domain we consider is an M-convex set, that is, the set of integral elements of an integral base-polyhedron. A fundamental difference between the fractional and the discrete case is that a…
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