Title:  The circuit diameter of polyhedra and related results

Relating the combinatorial diameter of a polyhedron to the number of its facets and its dimension remains a critical issue in optimization. In this talk we consider a variant of the combinatorial diameter called the circuit diameter, where walks are built using the circuit directions of the polyhedron. Under this framework, we study circuit versions of the Hirsch, nonrevisiting, and d-step conjectures, which are known to be equivalent for combinatorial diameter. We also discuss how simplicity of polyhedra and the wedge construction transfer to the circuit setting. Then we give a proof of the 4-step conjecture for circuits, and present some recent results in this area. Joint work with Steffen Borgwardt (CU Denver) and Tamon Stephen (SFU).