On the Greedy Algorithm for Combinatorial Auctions with a Random Order

TL;DR

This paper analyzes the greedy algorithm for combinatorial auctions with submodular bidders, showing that for vertex cover valuations and random item order, the expected approximation ratio improves from 2 to 1.75.

Contribution

It demonstrates an improved expected approximation ratio for the greedy algorithm under specific valuation functions and random ordering.

Findings

Expected approximation ratio is 7/4 for vertex cover valuations with random order.

The standard ratio of 2 is improved in this specific setting.

Random order assumption leads to better performance bounds.

Abstract

In this note we study the greedy algorithm for combinatorial auctions with submodular bidders. It is well known that this algorithm provides an approximation ratio of $2$ for every order of the items. We show that if the valuations are vertex cover functions and the order is random then the expected approximation ratio imrpoves to $\frac 7 4$.