# Approximation Schemes for a Unit-Demand Buyer with Independent Items via Symmetries

## TL;DR

This paper develops approximation schemes for revenue maximization in unit-demand settings with independent items, introducing symmetric menu complexity and a reduction technique to handle unbounded valuations efficiently.

## Contribution

It introduces the concept of symmetric menu complexity and provides a quasi-polynomial time mechanism that approximates optimal revenue for unit-demand buyers with independent items.

## Key findings

- A mechanism with quasi-polynomial symmetric menu complexity achieves near-optimal revenue.
- A polynomial-time reduction transforms unbounded valuation problems into bounded ones while preserving approximation guarantees.
- Selling items separately can be approximated with a linear symmetric menu complexity mechanism.

## Abstract

We consider a revenue-maximizing seller with $n$ items facing a single buyer. We introduce the notion of symmetric menu complexity of a mechanism, which counts the number of distinct options the buyer may purchase, up to permutations of the items. Our main result is that a mechanism of quasi-polynomial symmetric menu complexity suffices to guarantee a $(1-\varepsilon)$-approximation when the buyer is unit-demand over independent items, even when the value distribution is unbounded, and that this mechanism can be found in quasi-polynomial time. Our key technical result is a polynomial time, (symmetric) menu-complexity-preserving black-box reduction from achieving a $(1-\varepsilon)$-approximation for unbounded valuations that are subadditive over independent items to achieving a $(1-O(\varepsilon))$-approximation when the values are bounded (and still subadditive over independent items). We further apply this reduction to deduce approximation schemes for a suite of valuation classes beyond our main result. Finally, we show that selling separately (which has exponential menu complexity) can be approximated up to a $(1-\varepsilon)$ factor with a menu of efficient-linear $(f(\varepsilon) \cdot n)$ symmetric menu complexity.