To Trust or Not to Trust: Assignment Mechanisms with Predictions in the Private Graph Model

TL;DR

This paper designs strategyproof mechanisms that leverage predictions to achieve improved approximation guarantees for several variants of the Generalized Assignment Problem (GAP) in the private graph model, and draws inspiration from the renowned Gale-Shapley algorithm.

Abstract

The realm of algorithms with predictions has led to the development of several new algorithms that leverage (potentially erroneous) predictions to enhance their performance guarantees. The challenge is to devise algorithms that achieve optimal approximation guarantees as the prediction quality varies from perfect (consistency) to imperfect (robustness). This framework is particularly appealing in mechanism design contexts, where predictions might convey private information about the agents. In this paper, we design strategyproof mechanisms that leverage predictions to achieve improved approximation guarantees for several variants of the Generalized Assignment Problem (GAP) in the private graph model. In this model, first introduced by Dughmi & Ghosh (2010), the set of resources that an agent is compatible with is private information. For the Bipartite Matching Problem (BMP), we give a deterministic group-strategyproof (GSP) mechanism that is $(1 +1/γ)$-consistent and $(1 + γ)$-robust, where $γ\ge 1$ is some confidence parameter. We also prove that this is best possible. Remarkably, our mechanism draws inspiration from the renowned Gale-Shapley algorithm, incorporating predictions as a crucial element. Additionally, we give a randomized mechanism that is universally GSP and improves on the guarantees in expectation. The other GAP variants that we consider all make use of a unified greedy mechanism that adds edges to the assignment according to a specific order. Our universally GSP mechanism randomizes over the greedy mechanism, our mechanism for BMP and the predicted assignment, leading to $(1+3/γ)$-consistency and $(3+γ)$-robustness in expectation. All our mechanisms also provide more fine-grained approximation guarantees that interpolate between the consistency and the robustness, depending on some natural error measure of the prediction.

Figures (7)

• Figure 1: Limitations for $\text{GAP}\xspace$ with predictions.
• Figure 2: Taxonomy of GAP variants resulting from dughmi10 and our results. The respective color (column 1) indicates whether the problem is polynomial-time solvable (column 2) and whether an optimal algorithm gives rise to a strategyproof mechanism that is $1$-efficient (column 3).
• Figure 3: Instance used in the lower bound proof of Theorem \ref{['th:tradeOffConsisBoundedRobust']}.
• Figure 4: Impossibility trade-off in terms of $\alpha$-consistency and $\beta$-robustness. No deterministic strategyproof mechanism for $\text{BMP}$ can achieve a combination of $\alpha$ and $\beta$ in the gray area.
• Figure 5: Instance used in the lower bound proof of Theorem \ref{['th:lowerBoundWithError']}.

Theorems & Definitions (57)

• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.1
• Theorem 3.3
• proof
• Lemma 3.1: Lifting Lemma
• proof