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.

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

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

↑ 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, 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

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