Optimal assignments in an ordered set: An application of matroid theory

David Gale
1968 Journal of Combinatorial Theory  
Let X be a finite ordered set and let q~ be a function from X to subsets of a set Y. A subset A of X is called assignable if there is an injection ~b from A to Y that ~(a) ~ q~(a) for all a in A. It is shown that the assignable sets form a matroid on X and using this it is shown that there exists an optimal assignable set A, meaning that if A is any other assignable set then there is an injection f from A to A such thatf(a) > a for all a in A.
doi:10.1016/s0021-9800(68)80039-0 fatcat:kpmo3vilkbcnrnuusbfxioo7cu