Submodular Relaxation Methods

• Submodular Relaxation is a set of techniques that convert NP-hard discrete optimization problems into tractable continuous formulations using convex and polyhedral methods.
• These methods leverage continuous extensions such as the Lovász and multilinear extensions to enable efficient approximations in combinatorial and machine learning applications.
• Algorithmic frameworks like continuous greedy and rounding techniques provide near-optimal guarantees under constraints such as matroids, knapsacks, and energy minimization in graphical models.

Submodular relaxation refers to a family of techniques that relax discrete optimization problems involving submodular functions into continuous or convex optimization problems—often making them more tractable both theoretically and algorithmically. These relaxations underpin critical progress in both maximization and minimization, across domains including combinatorial optimization, machine learning, Markov random fields, and constraint satisfaction. The main conceptual mechanisms are continuous extensions (such as the multilinear and Lovász extensions), polyhedral relaxations, and generalized submodular (e.g., k-submodular) domains.

1. Fundamental Principles and Continuous Extensions

Submodular set functions $f:2^N \to \mathbb{R}$, characterized by the diminishing returns property, are highly expressive but the associated optimization problems are typically NP-hard. Submodular relaxation builds on the observation that submodular functions admit convex or concave continuous extensions which facilitate the use of continuous optimization tools.

The Lovász extension $\hat{f}:[0,1]^N\to\mathbb R$ is a canonical convex extension for submodular minimization. For $x\in[0,1]^N$, it integrates thresholded set evaluations: $$\hat{f}(x) = \int_0^1 f(\{i: x_i \ge \theta\})\,d\theta$$ It is convex if and only if $f$ is submodular, and exactly matches $f$ on indicator vectors. For maximization, the multilinear extension $F:[0,1]^N\to\mathbb{R}$ is employed: $F(x) = \mathbb{E}_{R \sim x}[f(R)]$ where $R$ is a random set including $i$ independently with probability $x_i$ (Chekuri et al., 2011).

Polyhedral relaxations include the convex hull of the discrete epigraph or outer approximations via polarity, yielding frameworks equivalent to or generalizing continuous extensions, and facilitating exact or approximate optimization (Atamturk et al., 2019).

2. Maximization: Multilinear Relaxation and Rounding

Problem Formulation and Relaxation

For maximizing $f(S)$ over $S \in \mathcal{I}$ (where $\mathcal{I}$ is a down-closed family), the original combinatorial problem is relaxed to: $\max_{x \in P} F(x)$ with $P = \mathrm{conv}\{1_S: S \in \mathcal{I}\}$, and $F$ the multilinear extension. Feasible solutions $x^*\in P$ are then rounded to a discrete set $S$ with small loss, using tools such as pipage or swap rounding (Chekuri et al., 2011, Buchbinder et al., 2024).

Approximability and Algorithms

The continuous greedy algorithm and its derivatives (e.g., measured continuous greedy for nonmonotone functions) achieve near-optimal approximation ratios. For a monotone $f$ and various constraints (matroid, knapsack, packing), $(1-e^{-1})$-approximation is achievable (Chekuri et al., 2018). For general nonmonotone $f$ under a matroid, a $0.309$-approximation is obtainable; this barrier is asymptotically tight due to symmetry-gap hardness (Vondrak, 2011). The addition of a modular term to $f$ (i.e., maximizing $f+\ell$) can be handled using adaptively weighted continuous greedy, with a parameterized approximation guarantee that interpolates between $1/e$ and $1$, depending on the sign structure of $\ell$ (Sun et al., 2022).

Recent advances include deterministic algorithms for the multilinear relaxation, using an “extended” multilinear extension in $2^{|N|}$ dimensions to avoid sampling bottlenecks and enable fully deterministic $(1/e-\epsilon)$-approximations under matroid constraints (Buchbinder et al., 2024).

Table: Key Maximization Guarantees via Submodular Relaxation

Constraint	Monotone $f$	Nonmonotone $f$	Reference
Matroid	$1-1/e$	$0.309$	(Chekuri et al., 2011)
Packing (Knapsack, etc.)	$1-1/e$	$0.25$–$0.309$	(Kulik et al., 2011)
Modular $+$ Non-monotone Submodular	$(1/e)$–$1$ (by β)	$0.478$ (lower bound)	(Sun et al., 2022)

3. Minimization: Lovász Extension and Polyhedral Methods

For submodular minimization: $\min_{S\subseteq N} f(S)$ the Lovász extension provides an exact convex relaxation: $\min_{x\in[0,1]^n} \hat{f}(x)$ Optimization over this convex surrogate admits efficient algorithms, and every minimizer of $\hat{f}$ can be rounded (often directly) to a minimizer of $f$ (Jaleel et al., 2017).

A polyhedral viewpoint employing the convex hull (epigraph) and its polarity yields a framework where cutting-plane algorithms (e.g., based on extended polymatroid inequalities) can be used; for submodular $f$, the polyhedral and continuous (Lovász) relaxations coincide (Atamturk et al., 2019).

For problems with additional constraints (e.g., linear constraints), extension frameworks (e.g., monotone closure reductions for two-variable-per-inequality IPs) admit tractable $2$-approximation algorithms; this ratio is tight under standard complexity assumptions (Hochbaum, 2010).

4. Convex Relaxations and Variants

Multiway Partition and Cost Allocation

For partition-type problems (e.g., Submodular Multiway Partition), convex programming via the Lovász extension leads to LP-equivalent relaxations: $$\min_x \sum_{i=1}^k \hat{f}_i(x^i) \quad \text{s.t.} \quad \sum_{i=1}^k x^i = 1_N$$ Achievable integrality gaps include $2$ for general submodular cost and $1.5-1/k$ when $f$ is symmetric (Chekuri et al., 2011, Mizutani, 1 Nov 2025).

Graphical Model Inference

Energy minimization in Markov random fields (MAP inference) benefits from submodular relaxation at the level of the Lagrangian dual. For pairwise (Potts) or specific high-order cases, the inner submodular energy minimization is computable via graph-cuts. The dual yields lower bounds competitive with LP relaxations, and subgradients or bundle methods can be efficiently applied (Osokin et al., 2015).

Learning and Structured Regularization

For learning graphical models with structured sparsity, convex programs using Lovász extensions of submodular priors (e.g., over edge sets for scale-free networks) yield tractable formulations with superior empirical properties relative to standard $\ell_1$ methods (Defazio et al., 2014).

Weak and $k$-submodular Relaxations

Generalizations (weak submodular, $k$-submodular) extend relaxation theory. Weak submodularity characterizes functions with a bounded violation of the diminishing returns inequality; relaxation-based greedy algorithms yield approximation ratios in terms of submodularity ratio parameters, with refined bounds attainable using local ratio analysis (Santiago et al., 2020). $k$-submodular relaxation extends the convexity–submodularity correspondence to labelings in $[0,k]^n$, with applications in constraint satisfaction (VCSPs) and FPT algorithms (Hirai et al., 2015).

5. Algorithmic Frameworks and Rounding

Solving the continuous relaxation typically requires either projection-free methods (continuous/local greedy, multiplicative weights) or LP/convex optimization within the feasible polytope. For maximization, rounding schemes such as pipage rounding, swap rounding, contention resolution schemes, or deterministic extended multilinear rounding guarantee small (often negligible) additional loss (Chekuri et al., 2011, Buchbinder et al., 2024). For minimization, subgradients can be computed greedily, and distributed consensus-based subgradient methods enable fully decentralized optimization (Jaleel et al., 2017).

Algorithmic developments for submodular relaxation also address deterministic vs. randomized query complexity—recent work has closed the gap for several classes of constraints and functions (Buchbinder et al., 2024).

6. Hardness and Tightness

Performance bounds for submodular relaxation are characterized by tight lower bounds:

• The multiway partition relaxation achieves (and cannot beat) $2$ (general) and $1.5-1/k$ (symmetric) integrality gaps (Chekuri et al., 2011).
• Nonmonotone submodular maximization under a matroid admits no better than $0.478$-approximation in polynomial time for general modular augmentation (Sun et al., 2022).
• Symmetry gap techniques show that for various constraints, the continuous relaxation and rounding framework is essentially tight in the value oracle model (Vondrak, 2011).

7. Applications and Broader Impact

Submodular relaxation frameworks are foundational in combinatorial optimization, network design, computer vision, machine learning, and distributed decision making. Notable applications include MAP inference in graphical models (Osokin et al., 2015), learning scale-free networks (Defazio et al., 2014), distributed agent coordination over lattices (Jaleel et al., 2017), and global optimization for value-based CSPs (Hirai et al., 2015). The architecture of submodular relaxations provides both theoretical guarantees and practical computational benefits, with ongoing generalizations continuing to expand their relevance in algorithmic and applied domains.