  We study the fair allocation of indivisible goods and chores under ordinal
valuations for agents with unequal entitlements. We show the existence and
polynomial time computation of weighted necessarily proportional up to one item
(WSD-PROP1) allocations for both goods and chores, by reducing it to a problem
of finding perfect matchings in a bipartite graph. We give a complete
characterization of these allocations as corner points of a perfect matching
polytope. Using this polytope, we can optimize over all allocations to find a
min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation
of chores. Additionally, we show the existence and computation of sequencible
(SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms
and show incompatibility of Pareto optimality under all valuations and
WSD-PROP1.
  We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our
characterization, we show the existence and polynomial time computation of
Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal
valuations for both chores and goods.
