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 $E$. Then define two associated polyhedra.

1. $EP_f:=\{x\in \mathbb{R}^E|\sum_{e\in U}x(e)\leq f(U), \forall U\subseteq E\}$
2. $P_f:=EP_f\cap \{x\in \mathbb{R}^E|x\geq 0\}$

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

Relation to matroids

If f is integer-valued, 1-Lipschitz, and $f(\emptyset)=0$ then f 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.

Properties

$P_f$ is nonempty if and only if $f\geq 0$ and that $EP_f$ is nonempty if and only if $f(\emptyset)\geq 0$.

Given any extended polymatroid $EP$ there is a unique submodular function $f$ such that $f(\emptyset)=0$ and $EP_f=EP$.

Contrapolymatroids

For a supermodular f one analogously may define the contrapolymatroid

$w \in\mathbb{R}_+^E: \forall S \subseteq E, \sum_{e\in S}w(e)\ge f(S)$

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

References

Footnotes

1. 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
2. Schrijver, Alexander (2003), Combinatorial Optimization, Springer, §44, p. 767, ISBN 3-540-44389-4

Additional reading

• Lee, Jon (2004), A First Course in Combinatorial Optimization, Cambridge University Press, ISBN 0-521-01012-8
• Fujishige, Saruto (2005), Submodular Functions and Optimization, Elsevier, ISBN 0-444-52086-4
• Narayanan, H. (1997), Submodular Functions and Electrical Networks, ISBN 0-444-82523-1