Covers and Packings in a Family of Sets by Jack Edmonds

1. For a finite set 5 of elements and a family F of subsets of S, a cover C of S from F is a subfamily CC.F such that U ( 0 = S. A cover C is called minimum if its cardinality \C\ is as small as possible. A packing D in F is a subfamily of i whose members are disjoint. I t is called maximum if its cardinality \D\ is as large as possible. Theorem 1 here is relevant to the task of finding minimum covers. Theorem 2, which follows easily from Theorem 1, is the analogous result on maximum packings. Finally, Theorem 3 extends the foregoing to "a-covers." Minimum covers are equivalent to solutions of the following integer program: Minimize ^Xi by a vector x = (xi, • • • , xn)' of zeroes and ones for which Ax^l = (l, • • • , 1)'. Here A is the zero-one incidence matrix of members of F (columns) versus members of S (rows). Where 1 is replaced by a vector a of arbitrary positive integers, Fulkerson and Ryser call min X)#* the ce-width of A. In [3] they find a lower bound for the a-width of zero-one matrices A with given row and column sums. By analogy with a-width, an a-cover of 5 from JF, where a is a vector whose components correspond to the members of S, is a subfamily of F of which at least ce* members contain yiÇzS. A /3-packing in F is a subfamily of F of which a t most ]8< members contain ^ £ S . Where Œi+fii is the number of members of F which contain yiÇzS, the complement in F of an a-cover is a /3-packing, and conversely. The Berge-Norman-Rabin theorem [ l ] , which concerns ce-covers where each member of F contains exactly two elements, is based on "alternating paths," invented in 1891 by Peterson and used frequently to prove theorems about linear graphs. Theorem 1 generalizes the N-R instance [5] of the B-N-R theorem, where a = ï , by extending the notion of alternating paths to "alternating t r e e s / Analogously, Theorem 2 includes the Berge theorem [2] on maximum matchings (packings of edges) in a graph. By introducing "selfintersecting trees," Theorem 3 generalizes the B-N-R theorem. Its proof, essentially the same as the proof of Theorem 1, is not given. Although present knowledge on practical algorithms is quite limited, in theory at least minimum covering and related program-