Support Rank & Submodular Partitioning

• Support Rank quantifies the complexity of set functions by decomposing them into simpler supermodular or submodular parts, enabling refined theoretical insights.
• Submodular Partitioning leverages these decompositions to split complex optimization tasks into tractable subproblems with improved approximation guarantees.
• The framework offers practical algorithms with balanced computational trade-offs and empirical performance gains, crucial for combinatorial optimization.

Support rank, as formalized by the notions of supermodular and submodular rank, quantitatively measures the complexity of set functions in terms of their decomposability into sums of "simpler" functions, specifically supermodular or submodular functions (with respect to varying partial orders). This structural perspective underpins both theoretical advances in set function analysis and algorithmic improvements for discrete optimization, particularly for functions that are not strictly submodular. Submodular partitioning leverages such decompositions to split optimization tasks on complex set functions into tractable subproblems, resulting in tighter approximation guarantees for a range of constrained combinatorial problems (Sonthalia et al., 2023).

1. Supermodular and Submodular Rank: Definitions and Bounds

Let $X$ be a finite lattice, such as $X = \{0,1\}^n$, and let $f: X \to \mathbb{R}$. A function is supermodular if, for all $x, y \in X$,

$f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$

Given a tuple of linear orders $\pi = (\pi_1, \ldots, \pi_n)$, a function is $\pi$-supermodular if this holds with respect to the product order $\le_\pi$. The convex cone of all $\pi$-supermodular functions is denoted $\mathcal{L}_\pi$.

The supermodular rank of $X = \{0,1\}^n$0 is the smallest integer $X = \{0,1\}^n$1 such that

$X = \{0,1\}^n$2

for some (possibly distinct) permutations $X = \{0,1\}^n$3.

For set functions $X = \{0,1\}^n$4, submodular rank is defined analogously as the supermodular rank of $X = \{0,1\}^n$5: $X = \{0,1\}^n$6

Structural Rank Bounds

Let $X = \{0,1\}^n$7 be the maximum supermodular rank for functions on $X = \{0,1\}^n$8 variables. For $X = \{0,1\}^n$9,

$f: X \to \mathbb{R}$0

The same upper bound applies to the submodular rank of set functions on $f: X \to \mathbb{R}$1 elements. Strictly submodular functions (interior of the submodular cone) attain rank exactly $f: X \to \mathbb{R}$2.

2. Elementary Submodular Rank and Decomposition

Rather than general partial orders, it suffices in practice to consider elementary partial orders—those differing from the identity by a flip in a single coordinate. The associated elementary submodular rank of $f: X \to \mathbb{R}$3 is the minimal $f: X \to \mathbb{R}$4 such that

$f: X \to \mathbb{R}$5

where $f: X \to \mathbb{R}$6 is submodular and each $f: X \to \mathbb{R}$7 lies in an elementary submodular cone corresponding to a coordinate flip.

A function has elementary submodular rank $f: X \to \mathbb{R}$8 if and only if there exists a set $f: X \to \mathbb{R}$9, $x, y \in X$0, such that the restriction

$x, y \in X$1

is submodular (as a function of $x, y \in X$2) for every $x, y \in X$3. This yields a decomposition into $x, y \in X$4 submodular "pieces," each corresponding to a specific assignment on $x, y \in X$5.

3. Decomposition and Partitioning Algorithm

Given $x, y \in X$6 and target elementary rank $x, y \in X$7, the decomposition procedure is as follows:

1. For each $x, y \in X$8 with $x, y \in X$9:
2. For every $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$0, restrict $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$1 to subsets $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$2 with $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$3, giving $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$4.
3. Verify if each $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$5 is submodular.
4. If all $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$6 are submodular, return a decomposition;
5. If not, increment $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$7 and repeat.

This method's correctness is established: it returns a valid decomposition if and only if the elementary submodular rank is at most $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$8, using exactly $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y).$9 elementary summands (Sonthalia et al., 2023).

4. Applications in Set Function Optimization

For a nonnegative monotone set-function $\pi = (\pi_1, \ldots, \pi_n)$0 on $\pi = (\pi_1, \ldots, \pi_n)$1, consider optimizing under cardinality or matroid constraints, or minimizing ratios $\pi = (\pi_1, \ldots, \pi_n)$2.

Submodular Partitioning for Maximization

Suppose $\pi = (\pi_1, \ldots, \pi_n)$3 has elementary submodular rank $\pi = (\pi_1, \ldots, \pi_n)$4 via decomposition over $\pi = (\pi_1, \ldots, \pi_n)$5, $\pi = (\pi_1, \ldots, \pi_n)$6: $\pi = (\pi_1, \ldots, \pi_n)$7 Each $\pi = (\pi_1, \ldots, \pi_n)$8 is submodular (on $\pi = (\pi_1, \ldots, \pi_n)$9), so any submodular maximization black-box algorithm $\pi$0 can be run independently on each piece. The best solution among the $\pi$1 outputs is then selected.

Approximation Guarantees

Let $\pi$2 denote the guarantee for $\pi$3 on submodular inputs. The "r–Split" scheme achieves: $\pi$4 where $\pi$5 (resp. $\pi$6) is the curvature for $\pi$7 (resp. best among the $\pi$8 partitions), and $\pi$9 is the submodularity ratio.

• Cardinality constraint with greedy yields

$\le_\pi$0

• Matroid constraint with continuous-greedy or pipage rounding yields the classical $\le_\pi$1 guarantee for a larger class of near-submodular functions.
• For $\le_\pi$2 ratio minimization, curvature-improved bounds also apply by using the best attainable $\le_\pi$3 over the subproblems.

Computational Trade-offs

For each $\le_\pi$4, partitioning produces $\le_\pi$5 subproblems, and there are $\le_\pi$6 potential $\le_\pi$7. The aggregate runtime is $\le_\pi$8, i.e., $\le_\pi$9. For fixed, small $\pi$0, this is practical, achieving polynomial-time approximation with increased accuracy at the cost of computational overhead.

5. Examples and Empirical Performance

Toy Example ($\pi$1, rank $\pi$2):

Let $\pi$3, and define

$\pi$4

$\pi$5 is not submodular globally but decomposes into two pieces depending on presence/absence of 3, each submodular in the restricted domain. Thus, elementary submodular rank is 2.

Running Greedy on the “must contain 3” and “must not contain 3” subproblems yields better guarantees than Greedy on $\pi$6 directly under cardinality constraint $\pi$7.

Practice and Empirics:

• Curvature-based bounds indicate that small curvature typically implies small submodular rank since large curvature violations force flips.
• Sampling-based tests may identify a candidate rank by empirically checking imset inequalities.
• Empirical evaluation (on determinantal, Bayesian-A-optimality, column-subset, and random functions up to $\pi$8) indicates that even for elementary ranks $\pi$9, 5–20% improvements over baseline Greedy are obtained with modest $\mathcal{L}_\pi$0 overhead (Sonthalia et al., 2023).

6. Connections with Submodular Optimization and Related Algorithms

The decomposition-and-partitioning framework systematically generalizes classical submodular optimization techniques, enabling their extension to a broader class of set functions by reducing tasks to parallel submodular subproblems.

Problem Type	Baseline Guarantee	Guarantee via r–Split
Cardinality, Greedy	$\mathcal{L}_\pi$1	$\mathcal{L}_\pi$2
Matroid, continuous	$\mathcal{L}_\pi$3	$\mathcal{L}_\pi$4 (for larger class)
Ratio $\mathcal{L}_\pi$5	$\mathcal{L}_\pi$6	Using minimized $\mathcal{L}_\pi$7

Classical references in this domain include work by Nemhauser, Wolsey & Fisher (1978), Calinescu et al. (2011), Bian et al. (2017), and Conforti & Cornuéjols (1984), with supermodular rank theory advanced by Montufar, Seigal & Uhler (Sonthalia et al., 2023).

7. Open Directions and Implications

Submodular rank introduces a fine-grained quantification of "distance from submodularity" in set functions, guiding both theoretical understanding and algorithm design. The practical decompositions offer a trade-off between solution quality and computational cost, parameterized by rank. A plausible implication is that many real-world optimization tasks—where set functions exhibit near-submodularity—may admit efficient decomposition leading to improved approximations at moderate overhead. The universality of rank-based partitioning raises prospects for extending submodular optimization paradigms to broader classes of constraints and set functions.