Interpolated Kneser-Ney is one of the best smoothing methods for n-gram language models. Previous explanations for its superiority have been based on intuitive and empirical justifications of specific properties of the method. We propose a novel interpretation of interpolated Kneser-Ney as approximate inference in a hierarchical Bayesian model consisting of Pitman-Yor processes. As opposed to p...

Kneser-Ney (1995) smoothing and its variants are generally recognized as having the best perplexity of any known method for estimating N-gram language models. Kneser-Ney smoothing, however, requires nonstandard N-gram counts for the lowerorder models used to smooth the highestorder model. For some applications, this makes Kneser-Ney smoothing inappropriate or inconvenient. In this paper, we int...

For a simple graph G = (V,E), the vertex boundary of a subset A ⊆ V consists of all vertices not in A that are adjacent to some vertex in A. The goal of the vertex isoperimetric problem is to determine the minimum boundary size of all vertex subsets of a given size. In particular, define μG(r) as the minimum boundary size of all vertex subsets of G of size r. Meanwhile, the vertex set of the Kn...

Abstract Let G be a graph. Assume that to each vertex of set vertices $S\subseteq V(G)$ robot is assigned. At stage one can move neighbouring vertex. Then S mobile general position if there exists sequence moves the robots such all are visited while maintaining property at times. The number cardinality largest . We give bounds on and determine exact values for certain common classes graphs, inc...

Recently we investigated in [SIAM J. Optim., 19 (2008), pp. 572–591] hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. In particular, we introduced two hierarchies of lower bounds: the “ψ”-hierarchy converging to the fractional chromatic number and the “Ψ”-hierarchy converging to the chromatic number of a graph. In both hierarchies the first order bounds are...

For each integer triple (n, k, s) such that k ≥ 2, s ≥ 2, and n ≥ ks, define a graph in the following manner. The vertex set consists of all k-subsets S of Zn such that any two elements in S are on circular distance at least s. Two vertices form an edge if and only if they are disjoint. For the special case s = 2, we get Schrijver’s stable Kneser graph. The general construction is due to Meunie...

A b-coloring of a graph G by k colors is a proper k-coloring of G such that in each color class there exists a vertex having neighbors in all the other k− 1 color classes. The b-chromatic number of a graph G, denoted by φ(G), is the maximum k for which G has a b-coloring by k colors. It is obvious that χ(G) ≤ φ(G). A graph G is b-continuous if for every k between χ(G) and φ(G) there is a b-colo...

The local boxicity of a graph G, denoted by lbox(G), is the minimum positive integer l such that G can be obtained using intersection k (where k≥l) interval graphs where each vertex appears as non-universal in at most these graphs. Let on n vertices having m edges. Δ denote maximum degree G. We show that, lbox(G)≤213log⁎⁡ΔΔ. lbox(G)∈O(nlog⁡n). lbox(G)≤(213log⁎⁡m+2)m. its product dimension. This...

This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels. A graph G is said to be d-distinguishable if there is a labeling of the vertex set using 1, . . . , d so that no nontrivial automorphism of G preserves the labels. A set of vertices S ⊆ V (G) is a determining set for G if every automorphism of G is...