Polymatroid

In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. ^[1] It is also described as the multiset analogue of the matroid.

Definition

Let ${\displaystyle E}$ be a finite set and ${\displaystyle f:2^{E}\rightarrow \mathbb {R} _{+}}$ a non-decreasing submodular function, that is, for each ${\displaystyle A\subseteq B\subseteq E}$ we have ${\displaystyle f(A)\leq f(B)}$, and for each ${\displaystyle A,B\subseteq E}$ we have ${\displaystyle f(A)+f(B)\geq f(A\cup B)+f(A\cap B)}$. We define the polymatroid associated to ${\displaystyle f}$ to be the following polytope:

${\displaystyle P_{f}={\Big \{}{\textbf {x}}\in \mathbb {R} _{+}^{E}~{\Big |}~\sum _{e\in U}{\textbf {x}}(e)\leq f(U),\forall U\subseteq E{\Big \}}}$.

When we allow the entries of ${\displaystyle {\textbf {x}}}$ to be negative we denote this polytope by ${\displaystyle EP_{f}}$, and call it the extended polymatroid associated to ${\displaystyle f}$. ^[2]

An equivalent definition

Let ${\displaystyle E}$ be a finite set. If ${\displaystyle {\textbf {u}},{\textbf {v}}\in \mathbb {R} ^{E}}$ then we denote by ${\displaystyle |{\textbf {u}}|}$ the sum of the entries of ${\displaystyle {\textbf {u}}}$, and write ${\displaystyle {\textbf {u}}\leq {\textbf {v}}}$ whenever ${\displaystyle {\textbf {v}}(i)-{\textbf {u}}(i)\geq 0}$ for every ${\displaystyle i\in E}$ (notice that this gives an order to ${\displaystyle \mathbb {R} _{+}^{E}}$). A polymatroid on the ground set ${\displaystyle E}$ is a nonempty compact subset ${\displaystyle P}$ in ${\displaystyle \mathbb {R} _{+}^{E}}$, the set of independent vectors, such that:

1. We have that if ${\displaystyle {\textbf {v}}\in P}$, then ${\displaystyle {\textbf {u}}\in P}$ for every ${\displaystyle {\textbf {u}}\leq {\textbf {v}}}$:
2. If ${\displaystyle {\textbf {u}},{\textbf {v}}\in P}$ with ${\displaystyle |{\textbf {v}}|>|{\textbf {u}}|}$, then there is a vector ${\displaystyle {\textbf {w}}\in P}$ such that ${\displaystyle {\textbf {u}}<{\textbf {w}}\leq (\max\{{\textbf {u}}(1),{\textbf {v}}(1)\},\dots ,\max\{{\textbf {u}}({|E|}),{\textbf {v}}({|E|})\})}$.

This definition is equivalent to the one described before, ^[3] where ${\displaystyle f}$ is the function defined by ${\displaystyle f(A)=\max {\Big \{}\sum _{i\in A}{\textbf {v}}(i)~{\Big |}~{\textbf {v}}\in P{\Big \}}}$ for every ${\displaystyle A\subset E}$.

Relation to matroids

To every matroid ${\displaystyle M}$ on the ground set ${\displaystyle E}$ we can associate the set ${\displaystyle V_{M}=\{{\textbf {w}}_{F}~|~F\in {\mathcal {I}}\}}$, where ${\displaystyle {\mathcal {I}}}$ is the set of independent sets of ${\displaystyle M}$ and we denote by ${\displaystyle {\textbf {w}}_{F}}$ the characteristic vector of ${\displaystyle F\subset E}$: for every ${\displaystyle i\in E}$

${\displaystyle {\textbf {w}}_{F}(i)={\begin{cases}1,&i\in F;\\0,&i\not \in F.\end{cases}}}$

By taking the convex hull of ${\displaystyle V_{M}}$ we get a polymatroid. It is associated to the rank function of ${\displaystyle M}$. The conditions of the second definition reflect the axioms for the independent sets of a matroid.

Relation to generalized permutahedra

Because generalized permutahedra can be constructed from submodular functions, and every generalized permutahedron has an associated submodular function, we have that there should be a correspondence between generalized permutahedra and polymatroids. In fact every polymatroid is a generalized permutahedron that has been translated to have a vertex in the origin. This result suggests that the combinatorial information of polymatroids is shared with generalized permutahedra.

Properties

${\displaystyle P_{f}}$ is nonempty if and only if ${\displaystyle f\geq 0}$ and that ${\displaystyle EP_{f}}$ is nonempty if and only if ${\displaystyle f(\emptyset )\geq 0}$.

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

Contrapolymatroids

For a supermodular f one analogously may define the contrapolymatroid

${\displaystyle {\Big \{}w\in \mathbb {R} _{+}^{E}~{\Big |}~\forall S\subseteq E,\sum _{e\in S}w(e)\geq f(S){\Big \}}}$

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

Discrete polymatroids

When we only focus on the lattice points of our polymatroids we get what is called, discrete polymatroids. Formally speaking, the definition of a discrete polymatroid goes exactly as the one for polymatroids except for where the vectors will live in, instead of ${\displaystyle \mathbb {R} _{+}^{E}}$ they will live in ${\displaystyle \mathbb {Z} _{+}^{E}}$. This combinatorial object is of great interest because of their relationship to monomial ideals.

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
3. J.Herzog, T.Hibi. Monomial Ideals. 2011. Graduate Texts in Mathematics 260, pp. 237–263 Springer-Verlag, London.

Additional reading

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