Showing papers in "Electronic Colloquium on Computational Complexity in 2017"
TL;DR: In this paper, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension was studied, and the best known upper bounds for both parameters were log(m), while the best lower bounds are linear in d.
Abstract: In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. Namely, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension. Let C be a binary concept class of size m and VC-dimension d. Prior to this work, the best known upper bounds for both parameters were log(m), while the best lower bounds are linear in d. We present significantly better upper bounds on both as follows. Set k = O(d2 d loglog | C | ).
19 citations
Journal Article•
TL;DR: In this paper, the complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication was studied and shown to be equivalent to the complement of polynomial identity testing (PIT).
Abstract: We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT). Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC ( ∪ , ∩ , x ‾ , + , × ) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ ( ∪ , ∩ , x ‾ , + , × ) studied by Glaser et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a well-studied, major open problem in algebraic computing complexity: • membership problem MC ( ∩ , + , × ) studied by McKenzie and Wagner 2007 • integer membership problems MC Z ( + , × ) , MC Z ( ∩ , + , × ) studied by Travers 2006 • equivalence problem EQ ( + , × ) studied by Glaser et al. 2010.
5 citations