Packing perfect matchings and dijoins

Consider a family of subsets F over a finite ground set T. A
(unweighted) packing problem is asking for the largest number of
pairwise disjoint sets in F. In cases where F has a nice enough
polyhedral description, one may use tools such as linear programming
duality and the theory of blocking polyhedra to derive min-max type
relations. Some examples of this include Menger?s theorem, max-flow
min-cut, etc. By relaxing integrality conditions on the coefficients
of the packing, one can look for fractional packings or packings where
we allow subtraction. Finally, the dyadic relaxations serve as a
middle step between lattice and linear programming relaxations, where
one can utilize both optimization tools and the algebraic structure of
dyadic rationals.In this talk, we consider two packing conjectures and
their lattice theoretical relaxations: one in the setting of perfect
matchings in a graph and another in the setting of dijoins and
strongly connected orientations in a digraph (time permitting). While
the two theories share many similarities in terms of polyhedral
descriptions and structural properties, there are still certain
distinctions leading to different lattice theoretical results. Based
on joint works with Ahmad Abdi, G\'erard Cornu\'ejols, and Siyue
Liu.