Results 1  10
of
16
Sphere Packing Numbers for Subsets of the Boolean nCube with Bounded VapnikChervonenkis Dimension
, 1992
"... : Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn ..."
Abstract

Cited by 94 (4 self)
 Add to MetaCart
: Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn=k) log(n=k)) d . This new result has applications in the theory of empirical processes. 1 The author gratefully acknowledges the support of the Mathematical Sciences Research Institute at UC Berkeley and ONR grant N0001491J1162. 1 1 Statement of Results Let n be natural number greater than zero. Let V ` f0; 1g n . For a sequence of indices I = (i 1 ; . . . ; i k ), with 1 i j n, let V j I denote the projection of V onto I, i.e. V j I = f(v i 1 ; . . . ; v i k ) : (v 1 ; . . . ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The VapnikChervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] (t...
Quantifying the Amount of Verboseness
, 1995
"... We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explic ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explicit description we consider several interesting combinatorial problems. 1 Introduction In the theory of bounded queries, we measure the complexity of a function by the number of queries to an oracle which are needed to compute it. The field has developed in various directions, both in complexity theory and in recursion theory; see Gasarch [21] for a recent survey. One of the original concerns is the classification of sets A of natural numbers by their "query complexity," i.e., according to the number of oracle queries that are needed to compute the nfold characteristic function F A n = x 1 ; : : : ; x n : (ØA (x 1 ); : : : ; ØA (x n )). In [3, 8] a set A is called verbose iff F A n is com...
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Defect Sauer Results
"... : In this paper we present a unified account of various results concerning traces of set systems, including the original lemma proved independently by Sauer [14], Shelah [15], and Vapnik and Chervonenkis [16], and extend these results in various directions. Included are a new criterion for a set sys ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
: In this paper we present a unified account of various results concerning traces of set systems, including the original lemma proved independently by Sauer [14], Shelah [15], and Vapnik and Chervonenkis [16], and extend these results in various directions. Included are a new criterion for a set system to be extremal for the Sauer inequality and upper and lower bounds, obtained by random methods, for the trace of a set system of size n r guaranteed on some ffn sized subset of f1; 2; : : : ; ng. 1 Notation A set system is a subset of P(n), the power set of [n] = f1; 2; : : : ; ng. The complement of a set I ae [n] is written I c , all other set differences are written out explicitly, e.g., P(n) n A. The system consisting of all sets of size k is written [n] (k) , while A (k) = A " [n] (k) , for any set system A. We define [n] (k) , A (k) , etc. similarly. Given A ae P(n) and I ae [n] we write I +A for fA [ I : A 2 Ag. Introduction A striking extremal result concerning ...
Witness sets
, 902
"... Abstract. Given a set C of binary ntuples and c ∈ C, how many bits of c suffice to distinguish it from the other elements in C? We shed new light on this old combinatorial problem and improve on previously known bounds. 1 ..."
Abstract
 Add to MetaCart
Abstract. Given a set C of binary ntuples and c ∈ C, how many bits of c suffice to distinguish it from the other elements in C? We shed new light on this old combinatorial problem and improve on previously known bounds. 1
On Minimum Saturated Matrices
, 2012
"... Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal satfunction for graphs, we introduce sattype problems for matrices. Let F be a family of krow matrices. A matrix M is called Fadmissible if M contains no submatrix F ∈ F (as a row and column ..."
Abstract
 Add to MetaCart
Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal satfunction for graphs, we introduce sattype problems for matrices. Let F be a family of krow matrices. A matrix M is called Fadmissible if M contains no submatrix F ∈ F (as a row and column permutation of F). A matrix M without repeated columns is Fsaturated if M is Fadmissible but the addition of any column not present in M violates this property. In this paper we consider the function sat(n, F) which is the minimal number of columns of an Fsaturated matrix with n rows. We establish the estimate sat(n, F) = O(n k−1) for any family F of krow matrices and also compute the satfunction for a few small forbidden matrices. 1
Locally identifying coloring of graphs
, 2012
"... We introduce the notion of locally identifying coloring of a graph. A proper vertexcoloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of u and v, respectively, ..."
Abstract
 Add to MetaCart
We introduce the notion of locally identifying coloring of a graph. A proper vertexcoloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of u and v, respectively, are distinct. Let χlid(G) be the minimum number of colors used in a locally identifying vertexcoloring of G. In this paper, we give several bounds on χlid for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether χlid(G) = 3 for a subcubic bipartite graph G with large girth is an NPcomplete problem.
Locally identifying coloring in bounded expansion classes of graphs
, 2013
"... A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent nontwin vertices are distinct. The lidchromatic number of a graph is the minimum number of colors used by a locally identifying vertexcoloring. In this paper, ..."
Abstract
 Add to MetaCart
A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent nontwin vertices are distinct. The lidchromatic number of a graph is the minimum number of colors used by a locally identifying vertexcoloring. In this paper, we prove that for any graph class of bounded expansion, the lidchromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lidchromatic number is bounded. This leads to an explicit upper bound for the lidchromatic number of planar graphs. This answers in a positive way a question of Esperet et al. [L. Esperet, S. Gravier, M. Montassier, P. Ochem and A. Parreau. Locally identifying coloring of graphs. Electronic Journal of Combinatorics, 19(2), 2012.].