首页/论文/2002.10925

arXiv 2002.109252020-02-22math.CO

A new solution to square matrix completion problem

Marija Dodig, Marko Stosic

备注：: 9 pages

英文摘要：

In this paper we give a novel solution to a classical completion problem for square matrices. This problem was studied by many authors through time, and it is completely solved in [2, 3]. In this paper we relate this classical problem to a purely combinatorial question involving partitions of integers and their majorizations studied in [4]. We show surprising relations in these approaches and as a corollary, we obtain a new combinatorial result on partitions of integers.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Sudoku Rectangle Completion

Over the last decade, Sudoku, a combinatorial number-placement puzzle, has become a favorite pastimes of many all around the world. In this puzzle, the task is to complete a partially filled $9 \times 9$ square with numbers 1 through 9, subject to the constraint that each number must appear once in each row, each column, and each of the nine $3 \times 3$ blocks. Sudoku squares can be considered a subclass of the well-studied class of Latin squares. In this paper, we study natural extensions of a classical result on Latin square completion to Sudoku squares. Furthermore, we use the procedure de…

2017-04-26math.CO

符号对称 $P_{0,1}^+$-矩阵补全问题

The Sign Symmetric $P_{0,1}^+$-Matrix Completion Problem

本文研究符号对称 $P_{0,1}^+$-矩阵补全问题，证明了非不对称不完全有向图缺乏该补全，完全分类了阶数不超过四的有向图，并与其他矩阵补全进行了比较。

2024-05-29math.CO

Completions of epsilon-dense partial Latin squares; quasirandom k-colorings of graphs

A classical question in combinatorics is the following:\ given a partial Latin square $P$, when can we complete $P$ to a Latin square $L$? In this paper, we investigate the class of \textbf{$ε$-dense partial Latin squares}:\ partial Latin squares in which each symbol, row, and column contains no more than $εn$-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Häggkvist conjectured that all $\frac{1}{4}$-dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a…

2013-06-03math.CO

Clique decompositions of multipartite graphs and completion of Latin squares

Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translate…

2017-04-24math.CO

Completions of epsilon-dense partial Latin squares

A classical question in combinatorics is the following: given a partial latin square P, when can we complete P to a latin square L? In this paper, we will investigate the class of \leqε-dense partial latin squares: partial latin squares in which each symbol, row, and column contains \leqεn-many nonblank cells. A conjecture of Nash-Williams on triangulations of graphs led Daykin and Häggkvist to conjecture that all \leq(1/4)-dense partial latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a …

2013-06-03math.CO

Realizability of Polytopes as a Low Rank Matrix Completion Problem

This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope.

2012-11-12math.CO

更多相关论文收起更多论文

The Signed Positive Semidefinite Matrix Completion Problem for Odd-$K_4$ Minor Free Signed Graphs

We give a signed generalization of Laurent's theorem that characterizes feasible positive semidefinite matrix completion problems in terms of metric polytopes. Based on this result, we give a characterization of the maximum rank completions of the signed positive semidefinite matrix completion problem for odd-$K_4$ minor free signed graphs. The analysis can also be used to bound the minimum rank over the completions and to characterize uniquely solvable completion problems for odd-$K_4$ minor free signed graphs. As a corollary we derive a characterization of the universal rigidity of odd-$K_4$…

2016-04-01math.COmath.OC

矩阵补全中的多个典型秩

Multiple typical ranks in matrix completion

本文研究实数域上矩阵补全的典型秩问题，证明多类指定元素集合只有一个典型秩，并刻画了具有两个典型秩的循环图族，完整确定了部分情形下的典型秩。

2025-03-12math.AGmath.CO

Restricted completion of sparse partial Latin squares

An $n \times n$ partial Latin square $P$ is called $α$-dense if each row and column has at most $αn$ non-empty cells and each symbol occurs at most $αn$ times in $P$. An $n \times n$ array $A$ where each cell contains a subset of $\{1,\dots, n\}$ is a $(βn, βn, βn)$-array if each symbol occurs at most $βn$ times in each row and column and each cell contains a set of size at most $βn$. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants $α, β> 0$ such that, for every positive integer $n$, if $P$ is an $α$-dense $n \times n$ partial La…

2016-08-26cs.DMmath.CO

矩阵补全判别式

Matrix Completion Discriminants

Tokyo University of Marine Science and Technology（东京海洋大学）；Tokyo University of Science（东京科学大学）

该研究定义了图的矩阵补全判别式，建立其团和公式与环的显式方程，结合K₀子式图分析证明可环补全图的判别式可分解为两类因子。

2026-10-16math.AGmath.COmath.OC

非负秩矩阵补全的几何结构

Geometry of low nonnegative rank matrix completion

该研究探讨了非负秩矩阵补全的几何结构，证明了非负秩1补全与秩1补全等价，但当r≥2时不成立，并通过几何特征分析了非负秩补全的条件。

2026-01-12math.AGmath.COmath.MG

双曲距离矩阵补全

Hyperbolic distance matrix completion

University of California at Santa Barbara（加州大学圣塔芭芭拉分校）；Université Laval（拉瓦尔大学）

本文在矩阵分析、图论与双曲几何交叉领域，提出双曲距离矩阵补全理论，利用锚定程序转化数据，以弦性为充要条件，并给出显式构造及应用。

2026-09-10math.COmath.MGmath.RA

Matrix Completion for the Independence Model

We investigate the problem of completing partial matrices to rank-one matrices in the standard simplex. The motivation for studying this problem comes from statistics: A lack of eligible completion can provide a falsification test for partial observations to come from the independence model. For each pattern of specified entries, we give equations and inequalities which are satisfied if and only if an eligible completion exists. We also describe the set of valid completions, and we optimize over this set.

2016-04-28math.AGmath.COmath.ST

Typical ranks in symmetric matrix completion

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks. We give a combinatorial description of the patterns of specified entires of $n\times n$ symmetric matrices that have $n$ as a typical rank. Moreover, we describe exactly when such a generic partial matrix is…

2020-10-14math.AGmath.COmath.ST

低秩矩阵补全的代数组合方法

The Algebraic Combinatorial Approach for Low-Rank Matrix Completion

Aalto University（阿尔托大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）

本文提出一种基于代数几何和拟阵理论的代数组合新视角，通过研究少量条目间的关系来解决低秩矩阵补全问题，并给出概率为1的算法判断特定条目是否可补全、从少量条目补全该条目及估计补全误差。

2014-08-19cs.LGcs.NAmath.AG