# counting quantifiers

## Vertex Cover Integrality Gap ★★

Author(s): Atserias

**Conjecture**For every there is such that, for every large , there are -vertex graphs and such that and .

Keywords: counting quantifiers; FMT12-LesHouches

## Fixed-point logic with counting ★★

Author(s): Blass

**Question**Can either of the following be expressed in fixed-point logic plus counting:

- \item Given a graph, does it have a perfect matching, i.e., a set of edges such that every vertex is incident to exactly one edge from ? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?

Keywords: Capturing PTime; counting quantifiers; Fixed-point logic; FMT03-Bedlewo