One Action Too Many: Inapproximability of Budgeted Combinatorial Contracts

TL;DR

This paper investigates the computational limits of designing multi-agent contracts with combinatorial actions under budget constraints, revealing strong inapproximability results and identifying tractable subclasses like gross substitutes.

Contribution

It establishes the first inapproximability results for submodular rewards in budgeted combinatorial contracts and provides a polynomial-time approximation for gross substitutes rewards.

Findings

No randomized polynomial-time algorithm can approximate submodular rewards within any finite factor under budget constraints.

Gross substitutes rewards admit a deterministic constant-factor approximation.

An FPTAS exists for additive rewards, even without budget constraints.

Abstract

We study multi-agent contract design with combinatorial actions, under budget constraints, and for a broad class of objective functions, including profit (principal's utility), reward, and welfare. Our first result is a strong impossibility: For submodular reward functions, no randomized poly-time algorithm can approximate the optimal budget-feasible value within \textit{any finite factor}, even with demand-oracle access. This result rules out extending known constant-factor guarantees from either (i) unbudgeted settings with combinatorial actions or (ii) budgeted settings with binary actions, to their combination. The hardness is tight: It holds even when all but one agent have binary actions and the remaining agent has just one additional action. On the positive side, we show that gross substitutes rewards (a well-studied strict subclass of submodular functions) admit a deterministic poly-time $O(1)$-approximation, using only value queries. Our results thus draw the first sharp separation between budgeted and unbudgeted settings in combinatorial contracts, and identifies gross substitutes as a tractable frontier for budgeted combinatorial contracts. Finally, we present an FPTAS for additive rewards, demonstrating that arbitrary approximation is tractable under any budget. This constitutes the first FPTAS for the multi-agent combinatorial-actions setting, even in the absence of budget constraints.