# On the Nisan-Ronen conjecture for submodular valuations

## TL;DR

This paper proves a fundamental lower bound on the approximation ratio for incentive compatible mechanisms in a scheduling domain with mostly unrelated machine valuations, extending the understanding of the Nisan-Ronen conjecture.

## Contribution

It establishes an $oldsymbol{ ext{Omega}( ext{sqrt}(n))}$ lower bound for deterministic mechanisms in a nearly unrelated machine scheduling domain with one submodular valuation, broadening previous results.

## Key findings

- Lower bound of $oldsymbol{ ext{Omega}( ext{sqrt}(n))}$ on approximation ratio.
- Impossibility result for incentive compatible mechanisms with expanded valuation classes.
- Novel characterization of smaller instances and linear mechanisms used in proof.

## Abstract

We consider incentive compatible mechanisms for a domain that is very close to the domain of scheduling $n$ unrelated machines: the single exception is that the valuation of just one machine is submodular. For the scheduling problem with such cost functions, we give a lower bound of $\Omega(\sqrt{n})$ on the approximation ratio of incentive compatible deterministic mechanisms. This is a strong information-theoretic impossibility result on the approximation ratio of mechanisms on relatively simple domains. The lower bound of the current work assumes no restriction on the mechanism side, but an expanded class of valuations, in contrast to previous general results on the Nisan-Ronen conjecture that hold for only special classes of mechanisms such as local, strongly monotone, and anonymous mechanisms. Our approach is based on a novel characterization of appropriately selected smaller instances that allows us to focus on particular type of algorithms (linear mechanisms), from which we extract a locality property that gives the lower bound.