Settling the Complexity Landscape of Multi-Agent Contracts with Binary Actions

2026-09-28

AI Summary

This study addresses the long-standing open problem regarding the computational complexity of optimal contract design over gross substitutes reward function classes in multi-agent binary action models. By conducting a fine-grained analysis of subclasses including OXS, WMRF, and partition rank, combined with approximation algorithm design and randomized query lower bound techniques, this work investigates the inherent tractability boundaries within the gross substitutes class. It reveals that submodularity serves as the critical factor for approximability. Specifically, the authors prove that OXS is APX-complete, establish an EPTAS for WMRF along with an FPTAS under specific settings, and demonstrate the strong inapproximability of ultra rewards. Collectively, these results provide a complete characterization of the complexity landscape for this problem, delineating the precise internal complexity boundaries within gross substitutes for the first time.

Abstract

We study the computational complexity of optimal contract design in the multi-agent binary-action model, focusing on gross-substitutes reward functions and related classes. While additive rewards admit an FPTAS and general submodular rewards admit only constant-factor approximation, the complexity within the intermediate class of gross substitutes has remained largely open. We uncover a fine-grained approximation landscape within this class. We first show that the optimal contract problem is APX-complete even for OXS rewards - a strict subclass of gross substitutes rewards, ruling out a PTAS for gross substitutes unless $\mathsf{P}=\mathsf{NP}$. In contrast, for weighted matroid rank functions (WMRFs) - another natural subclass of gross substitutes - we obtain an EPTAS and show that no randomized FPTAS exists in general. We further identify the special case of partition (weighted) matroid rank functions, for which we obtain an FPTAS. This stands in contrast to the multi-agent multi-action setting, where no PTAS exists even for unweighted partition matroid rank functions. Finally, we consider the broader class of ultra reward functions. While ultra rewards retain the tractability of gross substitutes in a related combinatorial contract model with a single agent, we show a sharp contrast in the multi-agent model: no polynomial-time randomized algorithm using value queries can achieve a $2^{o(n)}$-approximation in expectation. Together, our results reveal several qualitatively distinct computational regimes within and beyond gross substitutes, and identify submodularity as a crucial ingredient for the approximability of multi-agent contracts.

Problem

• multi-agent contracts
• computational complexity
• gross substitutes
• optimal contract design
• approximation

Innovation

• Multi-agent contracts
• Computational complexity
• Gross substitutes
• Approximation algorithms
• Matroid rank functions