A NOTE ON SUBMODULAR FUNCTIONS ON DISTRIBUTIVE LATTICES
分配束上の劣モジュラ関数について

Satoru Fujishige, Nobuaki Tomizawa
1983 Journal of the Operations Research Society of Japan  
Let D be a distributive lattice formed by subsets of a finite set E with 1/>, E E D and let R be the set of reals. Also let f be a submodular function from D into R with f(l/» = O. We determine the set of extreme points of the base polyhedron and give upper and lower bounds of f which can be obtained in polynomial time in IEI under mild assumptiOl\s. Without loss of generality we assume throughout the present paper that "each T. e: P of the poset P = (P,~) has cardinality one" ~ and we express
more » ... " ~ and we express P by (E") instead of (P") with P = {{e}le £ E}. Note that without the above assumption the base polyhedron B(f) does not have any extreme points. It should also be noted that because of this assumption both D(e) and D(e) -{e} belong to D for D(e) (e £ E) defined by (2.2) and that, for any integer i such that 0 ~ i ~ IEI, there exists a set X e: D with Ixl = i.
doi:10.15807/jorsj.26.309 fatcat:7tvi2hyktncpfh4ujh5matdnku