Algorithms as Mechanisms: The Price of Anarchy of Relax-and-Round

TL;DR

This paper explores how a broad class of approximation algorithms, when used as mechanisms with simple pricing rules, inherently possess low Price of Anarchy, extending previous results to more general algorithms and applications.

Contribution

It extends the connection between approximation algorithms and low Price of Anarchy to the relaxation-and-rounding design paradigm with smooth relaxations and oblivious rounding.

Findings

Mechanisms derived from relaxation-and-rounding algorithms have low Price of Anarchy.

The approach applies to sparse packing problems, TSP, and combinatorial auctions.

Results lead to simple, near-optimal mechanisms with improved guarantees.

Abstract

Many algorithms that are originally designed without explicitly considering incentive properties are later combined with simple pricing rules and used as mechanisms. The resulting mechanisms are often natural and simple to understand. But how good are these algorithms as mechanisms? Truthful reporting of valuations is typically not a dominant strategy (certainly not with a pay-your-bid, first-price rule, but it is likely not a good strategy even with a critical value, or second-price style rule either). Our goal is to show that a wide class of approximation algorithms yields this way mechanisms with low Price of Anarchy. The seminal result of Lucier and Borodin [SODA 2010] shows that combining a greedy algorithm that is an $\alpha$-approximation algorithm with a pay-your-bid payment rule yields a mechanism whose Price of Anarchy is $O(\alpha)$. In this paper we significantly extend the class of algorithms for which such a result is available by showing that this close connection between approximation ratio on the one hand and Price of Anarchy on the other also holds for the design principle of relaxation and rounding provided that the relaxation is smooth and the rounding is oblivious. We demonstrate the far-reaching consequences of our result by showing its implications for sparse packing integer programs, such as multi-unit auctions and generalized matching, for the maximum traveling salesman problem, for combinatorial auctions, and for single source unsplittable flow problems. In all these problems our approach leads to novel simple, near-optimal mechanisms whose Price of Anarchy either matches or beats the performance guarantees of known mechanisms.