Monotone cohomologies and oriented matchings

利用有向拟阵在非线性存在下寻找低秩结构

Using oriented matroids to find low rank structure in presence of nonlinearity

针对噪声与单调非线性失真下秩估计失效问题，提出利用有向拟阵理论计算潜在秩与单调秩，证明单调秩二判定为NP难，但引入有向拟阵完备化松弛问题使其可判定。

On the Holt-Klee Property for Oriented Matroid Programming

The Holt-Klee theorem says that the graph of a $d$-polytope, with edges oriented by a linear function on $P$ that is not constant on any edge, admits $d$ independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank $d+1$ on $n+2$ elements, which include those from $d$-polytopes with $n$ facets, admit $d$ independent monotone paths from source to sink if $d \le 4$. This was previously only known to hold for $d\le 3$ and $n\le 6$.

Monotone paths in random hypergraphs

We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs appear as line graph of uniform hypergraphs with vertex set $\mathbb N$.

A Proof of the Strict Monotone 5-step Conjecture

A computer search through the oriented matroid programs with dimension 5 and 10 facets shows that the maximum strictly monotone diameter is 5. Thus $Δ_{sm}(5,10)=5$. This enumeration is analogous to that of Bremner and Schewe for the non-monotone diameter of 6-polytopes with 12 facets. Similar enumerations show that $Δ_{sm}(4,9)=5$ and $Δ_m(4,9)=Δ_m(5,10)=6.$ We shorten the known non-computer proof of the strict monotone 4-step conjecture.

扫描、多胞体、定向拟阵与排列的容许图

Sweeps, polytopes, oriented matroids, and allowable graphs of permutations

本文引入扫描定向拟阵与排列的容许图两种推广，以建模高维点构型的扫描偏序集，揭示了其与模超平面、Dilworth截断及广义Baues问题的联系，并证明伪扫描偏序集形变收缩为球面。

Enumeration of Preferred Extensions in Almost Oriented Digraphs

In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on $n$ vertices, all preferred extensions can be enumerated in $O^*(3^{n/3})$ time and there are directed graphs with $Ω(3^{n/3})$ preferred extensions. We give faster enumeration algorithms for directed graphs with at most $0.8004\cdot n$ vertices occurring in $2$-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time $O(1.2321^n…