Ordered-Supermodularity: Theory and Applications

• Ordered-supermodularity is a property that extends classical supermodularity by integrating ordering structures, capturing complementarity and synergy in set systems.
• It underpins algorithmic strategies in discrete optimization, as seen in efficient greedy methods and Choquet integration over structured domains.
• Applications span probabilistic inequalities, sequential decision making, and economic models, providing actionable frameworks in research and practice.

Ordered-supermodularity is a concept that generalizes the classical notion of supermodularity by systematically incorporating ordering structures—whether over set systems, lattice products, or coordinate domains. It serves as a critical property in discrete optimization, probabilistic inequalities, cooperative game theory, sequential decision making, and the analysis of Markovian systems. Ordered-supermodularity captures synergy (complementarity) effects and monotonicity with respect to various orderings, thus enabling powerful structural and algorithmic results across these domains.

1. Foundational Concepts

The classical definition of a supermodular function for a set function $f: 2^N \to \mathbb{R}$ is

$$f(A) + f(B) \le f(A \cup B) + f(A \cap B), \qquad \forall A,B \subseteq N.$$

Supermodularity characterizes complementarity: adding an element to a large set increases the incremental value more than to a smaller one. Ordered-supermodularity arises when the domain is not just a Boolean algebra or lattice but is further endowed with some ordering—be it:

• an explicit ordering of items or indices,
• orderings on set systems (e.g., union-closed, intersection systems, or families with coverage/containment constraints),
• partial or total orders for coordinate tuples,
• restrictions induced by sequence or position in combinatorial or economic models.

In such situations, the classical submodular inequality is adapted to respect the natural ordering, leading to the analysis of more general or restrictive systems where only specific subsets, tuples, or orderings are meaningful, and where the function's value reflects this underlying structure.

2. Ordered Set Systems, Choquet Integration, and Ordered-Supermodularity

One influential extension of supermodularity to ordered families is given in the framework of the "Discrete Choquet Integral for Ordered Systems" (Faigle et al., 2011).

• Ordered system: A pair $(\mathcal{D}, \preceq)$, where $\mathcal{D}$ is a family of non-empty subsets of a finite ground set $N$ covering $N$, with a partial order $\preceq$.
• Choquet integral over ordered systems: For arbitrary functions on $\mathcal{D}$, the Choquet integral is defined via approximation by step functions correlating with $\mathcal{D}$'s order. The general Choquet integral is then expressed as both an infimum (over dominating linear functionals) and as a supremum (over submajorized step functions), taking into account only the order-structured feasible sets.
• Ordered-supermodularity: A set function (capacity) $v$ on $\mathcal{D}$ is supermodular with respect to the ordering if for all co-intersecting pairs $F, G \in \mathcal{D}$, there exist meet and join sets ($F \wedge G$, $F \vee G$), satisfying

$v(F \wedge G) + v(F \vee G) \ge v(F) + v(G)$

with the meet and join chosen compatibly with the order. This generalizes Lovász's result: in the classical case of the complete Boolean algebra, this coincides with the ordinary supermodularity.

• Algorithmic implications: For weakly union-closed or intersection systems, efficient greedy (Monge-type) algorithms correctly compute the Choquet integral, leveraging the ordered-supermodular property for correctness and tightness.

Applications are numerous: in cooperative games with restricted communication, in multi-criteria decision making on structured alternatives, and in combinatorial optimization where only certain flows/cuts are permitted.

3. Hierarchies, Decompositions, and Measures of Ordered-Supermodularity

Ordered-supermodularity in set functions can be quantified and classified through several recently developed hierarchies and decomposition theories:

3.1 MPH Hierarchy and Positive Lower Envelope (PLE) (Feige et al., 2014)

• Maximum over Positive Hypergraphs (MPH): Hierarchy $\mathcal{MPH}-k$ consists of monotone set functions expressible as the maximum over positive hypergraph functions with rank at most $k$. Monotone submodular functions coincide with $\mathcal{MPH}-1$.
• Supermodular degree: Captures functional "distance" from submodularity, and every function with supermodular degree $k$ lies in $\mathcal{MPH}-(k+1)$.
• Ordered-supermodular viewpoint: While not named as such, constructive procedures involve sequentially building PLEs, tracking marginal contributions respecting a chosen ordering. These arguments mirror ordered-supermodular behavior: monotonicity and "ordered" complementarity.

3.2 Supermodular Width and Related Hierarchies (Chen et al., 2018)

• Supermodular width (SMW): For a monotone function $f$, SMW(f) is the size of the largest subset witnessing supermodular complementarity. Functions with SMW $0$ are submodular; higher levels model "ordered" complementarity.
• Batched greedy maximization: Approximation guarantees for submodularity extend to SMW-d functions, scaling as $1 - e^{-1/(d+1)}$.
• Auction theory: The superadditive width (SAW) hierarchy further models intrinsic ordered complementarity relevant for efficiency guarantees in combinatorial auction mechanisms.

3.3 Decomposition and Rank (Sonthalia et al., 2023)

• Supermodular rank: Any set function can be written as a sum of ordered-supermodular summands (potentially with respect to different partial orders). The minimal number needed is the supermodular rank, measuring deviation from global supermodularity.
• Optimization: Algorithms exploit low rank by partitioning the domain so each restricted subproblem involves a submodular (ordered-supermodular) function, thus allowing known greedy or continuous methods to be applied with improved guarantees.

4. Ordered-Supermodularity in Probabilistic and Markovian Systems

Ordered-supermodularity also arises in stochastic analysis, especially via inequalities, correlation results, and process evolution.

• Multivariate stochastic orderings (Yin, 2015): A vector $X$ is less than $Y$ in supermodular order ($X \le_{SM} Y$) if $\mathbb{E}[f(X)] \le \mathbb{E}[f(Y)]$ for all supermodular functions $f$. Equality of expectations for aggregate convex (or concave) risk measures, combined with ordering, can force full distributional equality. This highlights the power of global ordered-supermodularity conditions.
• Generalized semimodularity and order statistics (Pinelis, 2019): Extends semimodularity and supermodularity to n-argument (lattice) functions, comparing the function's value at a general tuple to that at the tuple of order statistics—an explicit instance of "higher-dimensional" ordered-supermodularity. The central theorem (Theorem 1) shows that checking all pairwise (n:2) ordered-supermodular inequalities on a distributive lattice suffices to guarantee the full n-argument property.
• Monotonicity in Markov processes (Kella et al., 2020): For stochastically monotone Markov chains, expected values of supermodular functions of process coordinates exhibit monotonicity, convexity/concavity, or correlation decay as a function of time. The supermodular property, when applied to the tuple $(X_0, X_t)$ or difference variables, enables the systematic derivation of non-increasing covariance and other "ordered" regularity results.

5. Ordered-Supermodularity in Algorithmic and Economic Applications

5.1 Sequence and Position-Dependent Submodularity (Kleinberg et al., 2022)

• Ordered submodularity: For functions on sequences (ranked lists), a function is ordered-submodular if placing an item earlier yields at least as much marginal gain as placing it later, formalized as

$$f(A \Vert s) - f(A) \ge f(A \Vert s \Vert B) - f(A \Vert \bar{s} \Vert B)$$

for all prefixes $A$, candidates $s, \bar{s}$, and suffixes $B$. This models applications where order (e.g., prominence in recommendations) directly modulates utility. Ordered submodularity, although richer, yields weaker guarantees: greedy maximization yields a tight $2$-approximation (versus classical $1-1/e$).

5.2 Supermodularity in Norm Approximation (Kesselheim et al., 21 Jun 2024)

• $p$-supermodularity for norms: A (differentiable) norm $\|\cdot\|$ is $p$-supermodular if $u \mapsto \|u\|^p$ is supermodular on $\mathbb{R}^n_+$. The gradient and Hessian characterizations enforce the non-decreasing marginal cost property critical for smooth analysis and online algorithmic frameworks (e.g., load balancing, covering/packing problems, stochastic probing, bandits with knapsacks).
• Norm approximation: Every symmetric norm can be approximated by a $p$-supermodular norm, enabling the translation of techniques and analysis from classical $\ell_p$ spaces to broad algorithmic and online learning settings.

5.3 Ordered-Supermodularity in Game Theory (Yu, 30 Jun 2024)

• Order-theoretic relaxations: Conditions such as quasisupermodularity, meet- or join-subextremality, and the single crossing property generalize classical supermodularity and "increasing differences" to ordinal and lattice settings. Under such order-only assumptions, best response mappings in games remain monotone, and the set of Nash equilibria forms a complete lattice, extending Zhou's theorem purely order-theoretically rather than metrically.

6. Operations, Transformations, and Preservation of Ordered-Supermodularity

Canonical operations that preserve (ordered-)supermodularity are vital in function theory and algorithmic practice:

• Permutation, reflection, and lifting: If $m$ is ordered-supermodular, its image under permutations or set reflection maintains ordered-supermodularity. The lifting transformation embeds a function defined on a subset into a larger set, preserving the extremal structure.
• Product-based composition: Multiplying nondecreasing ordered-supermodular functions over disjoint variable blocks yields an ordered-supermodular function, and the extremality (atomicity) structure lifts accordingly.
• Projection and modular extension: Marginalizing or conditioning, or modular extension to a higher-dimensional space, commute with standardizations (imposing monotonicity/order), preserving both ordered-supermodularity and extremality.

Such transformation theory mirrors classical convex geometry and is foundational in areas such as Bayesian network learning and cooperative game theory.

7. Broader Implications and Emerging Directions

Ordered-supermodularity is a flexible and unifying paradigm. It underlies key equivalences in Choquet integration, guides the design of efficient greedy and batched optimization algorithms, governs probabilistic and stochastic inequalities, and structures the existence and uniqueness of equilibria in economic models with complex complementarities.

Emerging directions focus on further structural hierarchies (width, degree, rank), the orderly design of decomposition-based and scalable optimization techniques, sharper analysis of sequential and stochastic problems where order drives core performance metrics, and the nuanced paper of weaker order-based conditions sufficient for structure in large-scale systems.

In sum, ordered-supermodularity is an indispensable tool and theoretical backbone wherever discrete structure combines with order or complementarity, with substantial and diverse application in mathematics, computer science, economics, and operations research.