Greedy balanced pairs in $N$-free ordered sets

The 1/3-2/3 Conjecture for ordered sets whose cover graph is a forest

A balanced pair in an ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1/3, 2/3]$. We define the notion of a good pair and claim any ordered set that has a good pair will satisfy the conjecture and furthermore every ordered set which is not totally ordered and has a forest as its cover graph has a good pair.

Antimatroids and Balanced Pairs

We generalize the 1/3-2/3 conjecture from partially ordered sets to antimatroids: we conjecture that any antimatroid has a pair of elements x,y such that x has probability between 1/3 and 2/3 of appearing earlier than y in a uniformly random basic word of the antimatroid. We prove the conjecture for antimatroids of convex dimension two (the antimatroid-theoretic analogue of partial orders of width two), for antimatroids of height two, for antimatroids with an independent element, and for the perfect elimination antimatroids and node search antimatroids of several classes of graphs. A computer …

A greedy algorithm for the minimization of a ratio of same-index element sums from two positive arrays

Consider two ordered positive real number arrays of equal size. The problem is to find such set of indices of given size that the ratio of the sums of the array elements with those indices is minimized. In this work, in order to mitigate the exponential complexity of the brute force search, we present a greedy algorithm applied to the search of such an index set. The main result of the paper is the theorem that states that the algorithm eliminates from candidates all index sets that do not contain any elements from the greedily selected set. We additionally prove exactness for a particular cas…

On the $1/3-2/3$ Conjecture

Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we define $\mathbb{P}(x\prec y)$ to be the proportion of linear extensions of $P$ in which $x$ comes before $y$. For $0\leq α\leq \frac{1}{2}$, we say $(x,y)$ is an $α$-balanced pair if $α\leq \mathbb{P}(x\prec y) \leq 1-α.$ The $1/3-2/3$ Conjecture states that every finite partially ordered set which is not a chain has a $1/3$-balanced pair. We make progress on this conjecture by sh…

大型平衡独立集的尖锐在线困难性

Sharp Online Hardness for Large Balanced Independent Sets

研究稠密随机二分图中大型γ-平衡独立集的在线算法，证明最优规模为α，并给出达到(1-ε)(1-γ)α的在线算法及尖锐下界，揭示1/(1-γ)间隙的普遍性。

图的几乎平衡有序双团覆盖

Almost balanced ordered biclique covering of graphs

研究完全图K_n的最小双团覆盖大小f(n,k)，要求每条边被覆盖1到k次且有序计数平衡，对一般k给出了几乎紧的界。

Balanced paring of $\{1,2,\ldots,(p-1)/2\}$ for $p\equiv 1 \pmod{4}$

Let $p\equiv 1 \pmod{4}$ be a prime. Write $t = \prod_{x=1}^{(p-1)/2}x$. Since $t ^2\equiv -1 \pmod{p}$ , we can divide $\{1,2,\ldots,(p-1)/2\}$ into $(p-1)/4$ ordered pairs so that each pair, say $<a,\tilde{a}>$ , satisfies that $t a \equiv \pm \tilde{a} \pmod{p}.$ For any two such pairs, assume $a<\tilde{a}, b<\tilde{b}, a<b $, then there are three possibilities for their relative order : $a<\tilde{a} < b< \tilde{b}$ , $a< b < \tilde{a} < \tilde{b}$ , $a< b < \tilde{b}< \tilde{a}$. We show this paring is balanced in the sense that the three cases occur with equal frequencies. Utilizing prope…

自适应阈值驱动的连续贪心方法用于可扩展的子模优化

Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization

该研究提出了一种自适应阈值驱动的连续贪心方法（ATCG），用于解决在Matroid约束下的子模最大化问题，通过动态调整活跃集扩展策略，提高了算法效率并减少了通信开销。

Pattern-avoiding access in binary search trees

The dynamic optimality conjecture is perhaps the most fundamental open question about binary search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. one that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988; Munro, 2000; Demaine et al. 2009] [..] Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Theta(n log n), achievable by any balanced BST.…

On the Powers of Signed Graphs

A signed graph is an ordered pair $Σ=(G,σ),$ where $G=(V,E)$ is the underlying graph of $Σ$ with a signature function $σ:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an $n^{th}$ power of a signed graph to be unique. Also, we characterize balanced power signed graphs.

两个集合上的最终常数映射与Nilpotent对

Eventually constant maps for two sets and nilpotent pairs

本文通过双射关系，将两个集合上的最终常数映射与生成树联系起来，并扩展了Cayley公式和Leinster的工作，推导出nilpotent对数量的公式。