Polymatroid

In mathematics, polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970.^[1]

Definition

Consider any submodular set function $f$ on $S$ . Then define two associated polyhedra.

Here $P_f$ is called the polymatroid and $EP_f$ is called the extended polymatroid associated with $f$ ^[2].

$P_f$ is nonempty if and only if $f\geq 0$ and that $EP_f$ is nonempty if and only if $f(\phi)\geq 0$
Given any extended polymatroid $EP$ there is a unique submodular function $f$ such that $f(\phi)=0$ and $EP_f=EP$
If r is integral, 1-Lipschitz, and $r(\varnothing)=0$ then r is the rank-function of a matroid, and the polymatroid is the independent set polytope, so-called since Edmonds showed it is the convex hull of the characteristic vectors of all independent sets of the matroid.
For a supermodular f one analogously may define the contrapolymatroid

$w \in\mathbb{R}_%2B^E: \forall S \subseteq E, \sum_{e\in S}w(e)\ge r(S)$

This analogously generalizes the dominant of the spanning set polytope of matroids.

^ ^a ^b Edmonds, Jack. Submodular functions, matroids, and certain polyhedra. 1970. Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) pp. 69–87 Gordon and Breach, New York. MR 0270945
^ (Schrijver 2003, §44, p. 767)

Schrijver, Alexander (2003), Combinatorial Optimization, Springer, ISBN 3540443894
Lee, Jon (2004), A First Course in Combinatorial Optimization, Cambridge University Press, ISBN 0521010128
Fujishige, Saruto (2005), Submodular Functions and Optimization, Elsevier, ISBN 0444520864
Narayanan, H. (1997), Submodular Functions and Electrical Networks, ISBN 0444825231