Polymatroid Convolutions

• Polymatroid convolutions are operations that combine ranked lattices and additive measures to construct new polymatroid rank functions.
• They satisfy strict axioms—such as submodularity and monotonicity—that generalize classical convolution methods in matroid theory.
• Applications span discrete optimization, entropy geometry, and invariant construction, enabling effective extensions and non-stickiness analyses.

A polymatroid convolution is an operation that synthesizes new polymatroid rank functions from established mathematical structures, enabling both theoretical generalization and constructive design. Its significance spans discrete optimization, matroid theory, lattice theory, polyhedral geometry, and information theory. At its core, this convolution formalism links combinatorial lattices, discrete measures, and submodularity, offering a mechanism to encode extensions, amalgamation, structural invariants, and entropy-related properties within the framework of polymatroids.

1. Fundamental Definition and Construction

Let $M$ be a finite ground set. A ranked lattice $(\mathcal L,\,\rho)$ consists of a set $\mathcal L\subseteq 2^M$ closed under meet $(\wedge)$ and join $(\vee)$, with a monotone rank function $\rho:\mathcal L\to\mathbb R_{\ge 0}$ such that $\rho(\bot)=0$, where $\bot$ is the bottom element.

Let $\mu:M\to\mathbb R_{\ge 0}$ be a discrete additive measure, extended to subsets $A\subseteq M$ as $\mu(A)=\sum_{a\in A}\mu(a)$.

The polymatroid convolution, or lattice–measure convolution (Editor's term), is defined as

$$r(A) = (\rho*\mu)(A) = \min_{Z\in\mathcal L}\big\{\rho(Z) + \mu(A\setminus Z)\big\},\qquad A\subseteq M,$$

where $r:2^M\rightarrow\mathbb R_{\ge 0}$ is the resulting rank function. This construction generalizes classical polymatroid convolution—when $\mathcal L=2^M$ and $\rho$ is the rank of a polymatroid, this recovers $$(f*g)(A) = \min_{Y\subseteq A}\big\{f(Y) + g(A\setminus Y)\big\}$$ (Csirmaz, 2019, Matúš et al., 2013, Csirmaz, 2019).

2. Axiomatic Characterization and Polymatroidality

For $r(A)= (\rho*\mu)(A)$ to define a valid polymatroid rank, the following axioms on $(\mathcal L,\,\rho)$ and $\mu$ are necessary and sufficient (Csirmaz, 2019):

• (Z1) Pointedness: $\rho(\bot)=0$.
• (Z2*) Strict monotonicity: For $Z_1 < Z_2$ in $\mathcal L$, $0 < \rho(Z_2)-\rho(Z_1) < \mu(Z_2\setminus Z_1)$.
• (Z3) “Corrected” submodularity: For all $Z_1,Z_2\in \mathcal L$, $$\rho(Z_1)+\rho(Z_2) \ge \rho(Z_1\vee Z_2)+\rho(Z_1\wedge Z_2)+\mu((Z_1\cap Z_2)\setminus(Z_1\wedge Z_2))$$.
• (Z4) Atom–flat compatibility: For every $a\in M$ and $Z\in\mathcal L$ with $a\in Z$, $\mu(a)\le \rho(Z)$.
• (Z5) Non-degeneracy: $\rho(Z) > 0$ for $Z > \bot$; $\mu(a)>0$ for every $a\notin\bot$.

If (Z3) alone holds, $r$ is nonnegative, monotone, and submodular on $2^M$. If (Z1)-(Z5) all hold, $(\mathcal L, \rho)$ is exactly the lattice of cyclic flats with their ranks, and $\mu$ the singleton measure, of some polymatroid, and $r$ recovers the unique such polymatroid (Csirmaz, 2019, Csirmaz, 2019).

3. Connections to Matroids, Classical Convolution, and Generalizations

When $\mathcal L$ is the full boolean lattice and $\rho$ a polymatroid rank function, the lattice–measure convolution reduces to the classical convolution of two polymatroids (Matúš et al., 2013): $$(r_1*r_2)(X) = \min_{Y\subseteq X} \big\{r_1(Y) + r_2(X\setminus Y)\big\},$$ which is commutative and associative under standard conditions. In the matroid context, taking $\mu$ as the Boolean measure (0 on loops, 1 otherwise) and $\mathcal L$ as the lattice of cyclic flats recovers the Bonin–de Mier axioms and classical cyclic-flat matroid constructions. Earlier instances (e.g., Sims’ convolution-like embedding) are specializations to Boolean lattices (Csirmaz, 2019, Csirmaz, 2019).

The construction further generalizes: Starting from any finite ranked lattice and positive singleton measure produces a polymatroid with precisely that lattice of cyclic flats, significantly broadening constructive capability (Csirmaz, 2019).

4. Structural and Polyhedral Interpretations

The convolution admits a geometric realization in terms of lattice-point enumeration within Minkowski sums involving the base polytope of a polymatroid and scaled simplices. For a polymatroid $M=(E,r)$,

$$Q(M;u,t) = \#\big( (P_M + u\Delta + t\nabla)\cap \mathbb Z^E \big)$$

where $P_M$ is the base polytope, $\Delta$ the standard simplex, and $\nabla=-\Delta$. The convolution structure underpins the coefficient-wise and combinatorial properties of generalized Tutte-type polynomials and connects to activity expansions, polyhedral subdivisions, and modular decomposition (Cameron et al., 2016).

Context	Structure	Convolution Role
Matroids (cyclic flats)	Boolean lattice	Recovers classical convolution, cyclic flat axioms
General polymatroid construction	Arbitrary ranked lattice	Designs polymatroids via lattice and measure control
Tutte-like polynomial	Polytope with simplices	Minkowski-sum convolution for invariant construction

5. Applications: Extensions, Entropy, and Non-Stickiness

The convolution technique is essential in constructing polymatroid extensions, exploring amalgamation, and resolving the stickiness problem. Given two flats with a non-principal modular cut, convolution constructions realize one-point or two-point extensions that witness non-amalgamability by violating Ingleton-type inequalities, establishing that such polymatroids are not sticky (Csirmaz, 2019).

In information theory and entropy geometry, convolution enables direct-sum decompositions of the entropy region into tight and modular parts. The entropic cone closure is the direct sum of almost-entropic tight polymatroids and modular polymatroids. Computing convolutions corresponds to structural projections onto faces defined by balanced information inequalities, notably for Ingleton inequalities in four variables (Matúš et al., 2013).

6. Worked Examples and Computable Cases

For $M=\{a,b\}$, let $\mathcal L=\{\emptyset, Z\}$ with $Z=\{a,b\}$, assign $\rho(\emptyset)=0$, $\rho(Z)=1$, $\mu(a)=1$, $\mu(b)=1$. The convolution rank is:

• $r(\emptyset) = \min\{0+0,\,1+0\} = 0$
• $r(\{a\}) = \min\{0+1,\,1+0\} = 1$
• $r(\{b\}) = 1$
• $r(\{a,b\}) = \min\{0+2,\,1+0\} = 1$

This reconstructs the uniform matroid $U_{1,2}$ (Csirmaz, 2019, Csirmaz, 2019). More elaborate constructions (e.g., on chains, with tailored measures) are used to control local rank increments, violent modular cuts, and extension structure.

7. Perspectives and Generalizations

Polymatroid convolution serves as a unifying principle for manipulating submodular set functions, encoding combinatorial lattices, and constructing polymatroids with prescribed invariants. Extensions include convolution with two lattices, generalized permutohedra, Coxeter matroids, and connections to toric $K$-theory and the combinatorial geometry of Grassmannians (Cameron et al., 2016, Csirmaz, 2019). The lattice–measure framework suggests further research in modular decomposition, invariant theory, and the construction of counterexamples or obstructions in discrete geometric optimization.