Chapter 17. The Assignment Problem

Classical formulations of the assignment problem.

This page: Classical forms. View LLP forms »

This page collects the classical / sequential implementations of the algorithms developed in this chapter. The lattice-linear (LLP) reformulations and chapter setup live on the LLP companion page.

The assignment problem

Given $n$ bidders and $n$ items with valuations $v[b, i]$, find an assignment (perfect matching) and a price vector $C[1..n]$ such that every bidder is assigned to a most-preferred item at the given prices. Equivalently, find the minimum market clearing price vector: the smallest prices such that the preferred-items graph (edges to items maximising $v[b, i] - C[i]$) contains a perfect matching.

Overdemanded sets

A set $J$ of items is overdemanded if the set of bidders who prefer items in $J$ is strictly larger than $|J|$. By Hall's theorem, a perfect matching exists iff no overdemanded set exists. The LLP formulation uses overdemand as the forbidden predicate.