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Density Decomposition in Dual-Modular Optimization: Markets, Fairness, and Contracts

T-H. Hubert Chan, Shinuo Ma

TL;DR

The paper develops a unified dual-modular optimization framework with a density decomposition that yields canonical reward-to-cost densities and ties together market equilibrium, fairness notions, and combinatorial contracts. It shows that the density vector from decomposition coincides with densities induced by locally maximin allocations and equals the minimizer of a broad class of convex divergences, thus bridging density decomposition, lexicographic optimality, and contract design. It also introduces Frank-Wolfe-based iterative methods to approximate the density vector with convergence guarantees, and applies the framework to combinatorial contracts via dualities with Hockey-Stick divergences and DPI-based universality. Collectively, the results provide structural insights, algorithmic tools, and universal optimality principles for solving dual-modular problems across economics and combinatorial optimization.

Abstract

We study a unified framework for optimization problems defined on dual-modular instances, where the input comprises a finite ground set $V$ and two set functions: a monotone supermodular reward function $\f$ and a strictly monotone submodular cost function $\g$. This abstraction captures and generalizes classical models in economics and combinatorial optimization, including submodular utility allocation (SUA) markets and combinatorial contracts. At the core of our framework is the notion of density decomposition, which extends classical results to the dual-modular setting and uncovers structural insights into fairness and optimality. We show that the density decomposition yields a canonical vector of reward-to-cost ratios (densities) that simultaneously characterizes market equilibria, fair allocations -- via both lexicographic optimality and local maximin conditions -- and best-response strategies in contract design. Our main result proves the equivalence of these fairness notions and guarantees the existence of allocations that realize the decomposition densities. Our technical contributions include the analysis of a broad family of convex programs -- parameterized by divergences such as quadratic, logarithmic, and hockey-stick functions -- whose minimizers recover the density decomposition. We prove that any strictly convex divergence yields the same canonical density vector, and that locally maximin allocations act as universal minimizers for all divergences satisfying the data processing inequality. As an application of our framework, we determine the structure and number of critical values in the combinatorial contracts problem. Additionally, we generalize a Frank-Wolfe-type iterative method for approximating the dual-modular density decomposition, establishing both convergence guarantees and practical potential through efficient gradient oracle design.

Density Decomposition in Dual-Modular Optimization: Markets, Fairness, and Contracts

TL;DR

The paper develops a unified dual-modular optimization framework with a density decomposition that yields canonical reward-to-cost densities and ties together market equilibrium, fairness notions, and combinatorial contracts. It shows that the density vector from decomposition coincides with densities induced by locally maximin allocations and equals the minimizer of a broad class of convex divergences, thus bridging density decomposition, lexicographic optimality, and contract design. It also introduces Frank-Wolfe-based iterative methods to approximate the density vector with convergence guarantees, and applies the framework to combinatorial contracts via dualities with Hockey-Stick divergences and DPI-based universality. Collectively, the results provide structural insights, algorithmic tools, and universal optimality principles for solving dual-modular problems across economics and combinatorial optimization.

Abstract

We study a unified framework for optimization problems defined on dual-modular instances, where the input comprises a finite ground set and two set functions: a monotone supermodular reward function and a strictly monotone submodular cost function . This abstraction captures and generalizes classical models in economics and combinatorial optimization, including submodular utility allocation (SUA) markets and combinatorial contracts. At the core of our framework is the notion of density decomposition, which extends classical results to the dual-modular setting and uncovers structural insights into fairness and optimality. We show that the density decomposition yields a canonical vector of reward-to-cost ratios (densities) that simultaneously characterizes market equilibria, fair allocations -- via both lexicographic optimality and local maximin conditions -- and best-response strategies in contract design. Our main result proves the equivalence of these fairness notions and guarantees the existence of allocations that realize the decomposition densities. Our technical contributions include the analysis of a broad family of convex programs -- parameterized by divergences such as quadratic, logarithmic, and hockey-stick functions -- whose minimizers recover the density decomposition. We prove that any strictly convex divergence yields the same canonical density vector, and that locally maximin allocations act as universal minimizers for all divergences satisfying the data processing inequality. As an application of our framework, we determine the structure and number of critical values in the combinatorial contracts problem. Additionally, we generalize a Frank-Wolfe-type iterative method for approximating the dual-modular density decomposition, establishing both convergence guarantees and practical potential through efficient gradient oracle design.

Paper Structure

This paper contains 20 sections, 25 theorems, 51 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Dual-Modular Density Decomposition
  3. Analyzing Dual-Modular Density Decomposition, Markets and Contracts with Convex Program
  4. Iterative Approximation Methods
  5. Related Work
  6. Paper Organization
  7. Preliminaries
  8. Dual-Modular Density Decomposition
  9. Dual-Modular Market Allocation and Convex Program
  10. Iterative Approximation Methods
  11. Analysis of Dual-Modular Convex Program
  12. Structural Properties of Optimal Solutions
  13. An Example to Illustrate Strict Monotonicity Assumption
  14. Application to Dual-Modular Combinatorial Contracts
  15. Iterative Algorithms via Frank-Wolfe
  16. ...and 5 more sections

Key Result

Theorem 1.1

In a dual-modular instance $(V; \mathfrak{f}, \mathfrak{g})$, suppose $\rho^* \in \mathbb{R}^V$ is the density vector obtained from the density decomposition. Then, the following three conditions are equivalent for an allocation $(x, y) \in \mathcal{B}^{\geq}_\mathfrak{f} \times \mathcal{B}^{\leq}_\

Theorems & Definitions (57)

  • Theorem 1.1: Dual-Modular Density Decomposition and Market Fairness
  • Theorem 1.2: Density Decomposition Reveals Optimal Contract Response
  • Theorem 1.4: Equivalence Between Convex Program and Local Maximin Condition
  • Lemma 1.6: Duality Between Contracts and Hockey-Stick
  • Theorem 1.7: Universal Minimizer for Data-Processing Divergences
  • Theorem 1.8: Approximation Guarantees of Frank-Wolfe
  • Remark 1.9
  • Corollary 1.10: Symmetry Between Reward and Cost
  • Definition 2.1: Dual-Modular Input Instance
  • Definition 2.4: Dual-Modular Density Decomposition and Density Vector
  • ...and 47 more