Generalized Assignment Problem: Truthful Mechanism Design without Money

In this paper, we study a problem of truthful mechanism design for a strategic variant of the generalized assignment problem (GAP) in a both payment-free and prior-free environment. In GAP, a set of items has to be optimally assigned to a set of bins without exceeding the capacity of any singular bin. In the strategic variant of the problem we study, bins are held by strategic agents, and each agent may hide its compatibility with some items in order to obtain items of higher values. The compatibility between an agent and an item encodes the willingness of the agent to receive the item. Our goal is to maximize total value (sum of agents’ values, or social welfare) while certifying no agent can benefit from hiding its compatibility with items. The model has applications in auctions with budgeted bidders. For two variants of the problem, namely \emph{multiple knapsack problem} in which each item has the same size and value over bins, and \emph{density-invariant GAP} in which each item has the same value density over the bins, we propose truthful $4$-approximation algorithms. For the general problem, we propose an $O(\ln{(U/L)})$-approximation mechanism where $U$ and $L$ are the upper and lower bounds for value densities of the compatible item-bin pairs.

💡 Research Summary

The paper investigates truthful mechanism design without monetary transfers for a strategic variant of the Generalized Assignment Problem (GAP), termed GAP‑BS (Generalized Assignment Problem with Bin Strategic). In the classic GAP, a set of items must be assigned to a set of bins (knapsacks) without exceeding each bin’s capacity, maximizing total value. In GAP‑BS each bin is owned by a strategic agent who can hide a subset of the items it is compatible with; the hidden edges represent items the agent pretends it cannot receive. The public data consist of the value matrix v_{ij}, size matrix w_{ij}, and bin capacities C_i, while each agent’s private type is the set of compatible edges E_i. The goal is to design a mechanism that (i) always produces a feasible assignment, and (ii) is dominant‑strategy incentive compatible (DSIC) – no bin can increase its expected value by misreporting a smaller compatibility set.

The authors adopt the welfare‑approximation framework: a mechanism is α‑approximate if its expected total value is at least 1/α of the optimal (offline) GAP optimum. This relaxation is necessary because Gibbard‑Satterthwaite‑type impossibility results forbid exact optimal truthful mechanisms without money.

Technical Approach

1. LP Relaxation – The problem is first expressed as a linear program (LP