Orthant Submodular Function

Updated 5 November 2025

Orthant submodular functions are defined as those that exhibit submodular or supermodular behavior within each orthant of a product order, generalizing set-based submodularity.
They underpin the theory of k-submodular and bisubmodular functions and allow decomposition into sums with low supermodular/submodular rank, facilitating specialized optimization algorithms.
Algorithmic approaches leveraging orthant submodularity include greedy and randomized methods, with practical implications in discrete optimization, lattice structures, and continuous domain extensions.

An orthant submodular function is a generalization of classical submodularity in which the function is required to exhibit submodular structure within each orthant or, more generally, with respect to distinct product orders on the ground set. This property is central in discrete optimization, especially in the paper of $k$ -submodular and set-valued functions, function decompositions, and a range of maximization and minimization algorithms that leverage the richer structure beyond standard (set-based) submodularity.

1. Formal Definitions and Fundamental Properties

Let $X = \{0,1\}^n$ denote the Boolean cube, or more generally, a product domain with $n$ coordinates taking values in a fixed set. The notion of orthant submodularity is defined with respect to a tuple of linear orders $\pi = (\pi_1,\dots,\pi_n)$ , where each $\pi_i$ is a total order on coordinate $i$ . For each $\pi$ , the induced cone of $\pi$ -supermodular (or, equivalently, $\pi$ -submodular) functions consists of those $f: X \to \mathbb{R}$ satisfying

$f(x) + f(y) \leq f(x \wedge_\pi y) + f(x \vee_\pi y) \qquad \forall x, y \in X$

for appropriate meet and join operations parameterized by $\pi$ . Orthant submodular functions are precisely those functions that are submodular (or supermodular) with respect to possibly distinct product orders; this class strictly generalizes classical submodular and bisubmodular functions (Sonthalia et al., 2023, Ward et al., 2014).

A key structural result is that the set of all functions of low "supermodular rank" are precisely those that can be written as sums of such orthant submodular functions, each possibly associated with a different product order.

2. Orthant Submodularity in the Theory of $k$ -Submodular and Bisubmodular Functions

Orthant submodular functions play a foundational role in the extension of submodularity from sets ( $k=1$ ) to signed or colored assignments ( $k\geq 2$ ). In the $k$ -submodular framework, functions $f: \{0, \ldots, k\}^U \to \mathbb{R}$ are called $k$ -submodular if for all $s, t$ : $f(s) + f(t) \geq f(\min_0(s, t)) + f(\max_0(s, t))$ Here, the operation $\min_0$ and $\max_0$ act componentwise, preserving the structure of orthants where each orthant represents a fixed assignment of each element to a label or unassigned. A function is said to be submodular in every orthant if, when restricted to any orthant (full assignment), it satisfies the classical submodularity inequality.

A central theorem links orthant submodularity (submodularity in every orthant) with "pairwise monotonicity" to characterize $k$ -submodular functions: $f$ is $k$ -submodular if and only if it is submodular in every orthant and pairwise monotone (Ward et al., 2014).

For $k=2$ , orthant submodular functions coincide with bisubmodular functions, which are precisely those set functions defined on $\{0,+,-\}^n$ satisfying the necessary submodularity inequalities within each orthant.

3. Decomposition via Supermodular/Submodular Rank

The structure theory articulated in (Sonthalia et al., 2023) identifies orthant submodular functions as the algebraic building blocks for arbitrary set functions. Any $f: \{0,1\}^n \to \mathbb{R}$ can be decomposed as a sum

$f = f_1 + \cdots + f_r$

where each $f_i$ is $\pi^{(i)}$ -supermodular with respect to a possibly different order $\pi^{(i)}$ . The minimal such $r$ is the supermodular rank, and analogously, submodular rank is defined for decomposition into orthant submodular functions (in the sense of submodularity).

Key results:

The maximum possible supermodular rank for discrete set functions is $\lceil \log_2 n \rceil + 1$ for $n \ge 3$ .
The maximum elementary submodular rank is $n$ .
Decomposition into sums of orthant submodular functions (with suitable product orders) enables partition of the original domain into $2^r$ pieces, each supporting classical submodularity, which is algorithmically exploitable.

4. Algorithmic Implications and Optimization Methods

Orthant submodular structure underpins both algorithmic maximization and minimization strategies. Several pivotal frameworks exploit this:

Maximization via Greedy Algorithms: For $k$ -submodular functions (characterized as orthant submodular and pairwise monotone), deterministic greedy algorithms yield a $1/(1+r)$-approximation for the broader class of functions submodular in every orthant and $r$ -wise monotone. In particular, $1/3$-approximation is achieved for maximizing $k$ -submodular functions, matching the best possible known for deterministic algorithms in the value oracle model (Ward et al., 2014).
Randomized Greedy Methods: For functions that are submodular in every orthant and $k$ -wise monotone, randomized greedy algorithms obtain an approximation ratio of $1/(1+\sqrt{k})$ ; for $k$ -submodular functions, the guarantee is at least $1/(1+\max(1, \sqrt{k-1}))$ .
Decomposition-Guided Maximization: The $r$ -Split algorithm, given a decomposition of $f$ into $r+1$ elementary orthant submodular components, partitions the optimization problem into $2^r$ subproblems, each over genuinely submodular pieces. Given that state-of-the-art submodular maximization admits strong approximation ratios, the overall guarantee is given by

$\max\{ R(\alpha, \gamma), R(\alpha_r, 1) \}$

where $R$ is the approximation ratio as a function of curvature and submodularity ratio parameters (Sonthalia et al., 2023).

Minimization Complexity: An exact polynomial-time combinatorial minimization algorithm is known for orthant submodular (bisubmodular, $k=2$ ) functions; for $k$ -submodular functions ( $k>2$ ), despite the Min-Max-Theorem, no pseudopolynomial algorithm is known (Huber et al., 2013, Ward et al., 2014). The lack of a “colored convexity” structure, in contrast to orthant submodular functions, is a key obstruction.

5. Extension to Lattices and Continuous Domains

Orthant submodular structure generalizes to integer lattices and distributive lattices, where the meet and join operations are defined coordinatewise in the corresponding partial orders. For $f: \mathbb{Z}^n \to \mathbb{R}$ (integer lattice), submodularity (sometimes called "orthant submodularity") is defined as: $f(x) + f(y) \ge f(x \land y) + f(x \lor y) \qquad \forall x, y$ However, in these non-Boolean settings, orthant submodularity is strictly weaker than the Diminishing Returns (DR) property. On the integer or distributive lattice, DR-submodularity is necessary for approximation algorithms matching set-function guarantees; orthant submodularity without DR only supports much weaker or no guarantees—specifically, the deterministic double greedy algorithm achieves only a $1/3$-approximation on the integer lattice in the absence of DR (Gottschalk et al., 2015).

In the continuous setting, entropy-like (EL) functions defined on the non-negative orthant $\mathbb{R}^n_{\ge 0}$ —those that are submodular, monotone, pointed, and satisfy diminishing returns—are salient for optimization problems under geometric constraints, further developing the interplay between orthant submodularity and analytic function theory (Csirmaz, 2020).

6. Structural and Practical Relevance

Orthant submodular functions and their decompositions offer a unifying framework that interpolates between fully modular functions (rank 1) and arbitrary set functions. Many functions arising in practice (e.g., in probabilistic graphical models, secret sharing schemes, and certain VCSP relaxations) are not globally submodular but can be represented as sums of a small number of orthant submodular components. The concept of supermodular (or submodular) rank provides a precise grading for this structure.

Optimization strategies can then exploit knowledge of supermodular/submodular rank, moving smoothly between algorithms for general non-submodular functions and those with guaranteed submodular structure, and improving approximation ratios as the rank decreases. Theoretical connections extend to mixtures and latent variable models, as the log-probabilities associated with them correspond to functions of low supermodular rank (Sonthalia et al., 2023).

Table: Characterization and Key Properties

Concept	Definition	Algorithmic/Structural Significance
Orthant submodular	Submodular (or supermodular) w.r.t. some product order $\pi$	Building block of general set functions
Supermodular rank	Minimum $r$ with $f = \sum_{i=1}^r$ orthant supermodular $f_i$	Interpolates modular to arbitrary
Submodular rank	Minimum $r$ with $f = \sum_{i=1}^r$ orthant submodular $f_i$	Decomposition-driven optimization
$k$ -submodular	Submodular in every orthant, pairwise monotone	Generalizes submodular, bisubmodular
DR-submodular	Diminishing returns in lattice domain	Enables strong maximization guarantees

7. Open Questions and Future Directions

Major open problems include the design of efficient minimization algorithms for $k$ -submodular functions with $k>2$ (the Min-Max-Theorem provides a dual characterization but no known practical algorithm), the extension of strong approximability or tractability to broader classes within the orthant submodular framework, the tightness of lower bounds for continuous entropy-like functions under general geometric constraints, and the operational meaning of submodular/supermodular rank in high-dimensional probabilistic modeling (Huber et al., 2013, Ward et al., 2014, Sonthalia et al., 2023, Csirmaz, 2020). These ongoing questions continue to drive research at the intersection of combinatorial and continuous submodularity, discrete convexity, and algorithmic optimization.