亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

A simple and effective method for the alignment of generative models is the best-of-$n$ policy, where $n$ samples are drawn from a base policy, and ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-$n$ policy and the base policy is equal to $\log (n) - (n-1)/n.$ We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes. Finally, we propose a new estimator for the KL divergence and empirically show that it provides a tight approximation through a few examples.

For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers $k\geq 1$, $s\geq 2$, and $n \geq sk+1$, the $s$-stable Kneser graph $S(n,k,s)$ has as vertices all $k$-element subsets of $[n]$ in which any two elements are in cyclical distance at least $s$. We prove that all the graphs $S(n,k,s)$, in particular Schrijver graphs $S(n,k)=S(n,k,2)$, admit a Hamilton cycle that can be computed in time $\mathcal{O}(n)$ per generated vertex.

Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical binomial random graph $G_{n,p}$ has been analysed and shown to have largest independent sets of size $\Theta(\log{n})$ w.h.p. This classical model does not capture any dependency structure between edges that can appear in real-world networks. We initiate study in this direction by defining random graphs $G^{r}_{n,p}$ whose existence of edges is determined by a Markov process that is also governed by a decay parameter $r\in(0,1]$. We prove that w.h.p. $G^{r}_{n,p}$ has independent sets of size $(\frac{1-r}{2+\epsilon}) \frac{n}{\log{n}}$ for arbitrary $\epsilon > 0$, which implies an asymptotic lower bound of $\Omega(\pi(n))$ where $\pi(n)$ is the prime-counting function. This is derived using bounds on the terms of a harmonic series, Tur\'an bound on stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Since $G^{r}_{n,p}$ collapses to $G_{n,p}$ when there is no decay, it follows that having even the slightest bit of dependency (any $r < 1$) in the random graph construction leads to the presence of large independent sets and thus our random model has a phase transition at its boundary value of $r=1$. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most $1 + \frac{\log{n}}{(1-r)}$ w.h.p. when the lowest degree vertex is picked at each iteration, and also show that under any other permutation of vertices the algorithm outputs a set of size $\Omega(n^{1/1+\tau})$, where $\tau=1/(1-r)$, and hence has a performance ratio of $O(n^{\frac{1}{2-r}})$.

A tuple (Z_1,...,Z_p) of matrices of size r is said to be a commuting extension of a tuple (A_1,...,A_p) of matrices of size n <r if the Z_i pairwise commute and each A_i sits in the upper left corner of a block decomposition of Z_i. This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called "quantum Zeno dynamics." Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: (i) Theorems on the uniqueness of commuting extensions for three matrices or more. (ii) Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1. (iii) A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $\ell_1$ and $\ell_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In the the bipartite permutation vertex deletion problem we ask for a given $n$-vertex graph, whether we can remove at most $k$ vertices to obtain a bipartite permutation graph. This problem is NP-complete but it does admit an FPT algorithm parameterized by $k$. In this paper we study the kernelization of this problem and show that it admits a polynomial kernel with $O(k^{62})$ vertices.

Fix a positive integer $n$, a real number $p\in (0,1]$, and a (perhaps random) hypergraph $\mathcal{H}$ on $[n]$. We introduce and investigate the following random multigraph model, which we denote $\mathbb{G}(n,p\, ; \,\mathcal{H})$: begin with an empty graph on $n$ vertices, which are labelled by the set $[n]$. For every $H\in \mathcal{H}$ choose, independently from previous choices, a doubleton from $H$, say $D = \{i,j\} \subset H$, uniformly at random and then introduce an edge between the vertices $i$ and $j$ in the graph with probability $p$, where each edge is introduced independently of all other edges.

A subset $I$ of the vertex set $V(G)$ of a graph $G$ is called a $k$-clique independent set of $G$ if no $k$ vertices in $I$ form a $k$-clique of $G$. An independent set is a $2$-clique independent set. Let $\pi_k(G)$ denote the number of $k$-cliques of $G$. For a function $w: V(G) \rightarrow \{0, 1, 2, \dots\}$, let $G(w)$ be the graph obtained from $G$ by replacing each vertex $v$ by a $w(v)$-clique $K^v$ and making each vertex of $K^u$ adjacent to each vertex of $K^v$ for each edge $\{u,v\}$ of $G$. For an integer $m \geq 1$, consider any $w$ with $\sum_{v \in V(G)} w(v) = m$. For $U \subseteq V(G)$, we say that $w$ is uniform on $U$ if $w(v) = 0$ for each $v \in V(G) \setminus U$ and, for each $u \in U$, $w(u) = \left\lfloor m/|U| \right\rfloor$ or $w(u) = \left\lceil m/|U| \right\rceil$. Katona asked if $\pi_k(G(w))$ is smallest when $w$ is uniform on a largest $k$-clique independent set of $G$. He placed particular emphasis on the Sperner graph $B_n$, given by $V(B_n) = \{X \colon X \subseteq \{1, \dots, n\}\}$ and $E(B_n) = \{\{X,Y\} \colon X \subsetneq Y \in V(B_n)\}$. He provided an affirmative answer for $k = 2$ (and any $G$). We determine graphs for which the answer is negative for every $k \geq 3$. These include $B_n$ for $n \geq 2$. Generalizing Sperner's Theorem and a recent result of Qian, Engel and Xu, we show that $\pi_k(B_n(w))$ is smallest when $w$ is uniform on a largest independent set of $B_n$. We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs.

For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.

We prove the following variant of Levi's Enlargement Lemma: for an arbitrary arrangement $\mathcal{A}$ of $x$-monotone pseudosegments in the plane and a pair of points $a,b$ with distinct $x$-coordinates and not on the same pseudosegment, there exists a simple $x$-monotone curve with endpoints $a,b$ that intersects every curve of $\mathcal{A}$ at most once. As a consequence, every simple monotone drawing of a graph can be extended to a simple monotone drawing of a complete graph. We also show that extending an arrangement of cylindrically monotone pseudosegments is not always possible; in fact, the corresponding decision problem is NP-hard.

Let the costs $C(i,j)$ for an instance of the asymmetric traveling salesperson problem be independent uniform $[0,1]$ random variables. We consider the efficiency of branch and bound algorithms that use the assignment relaxation as a lower bound. We show that w.h.p. the number of steps taken in any such branch and bound algorithm is $e^{\Omega(n^a)}$ for some small absolute constant $a>0$.