Greediness is not always a vice: Efficient Discovery Algorithms for Assignment Problems

TL;DR

This paper introduces efficient greedy discovery algorithms for the assignment problem that minimize weight queries while guaranteeing solution quality, motivated by applications in energy sharing communities.

Contribution

It presents novel greedy algorithms for the discovery variant of the assignment problem, addressing inherent challenges and leveraging natural node processing order assumptions.

Findings

Algorithms reduce the number of queried weights significantly.

Guarantees on solution quality are maintained despite limited information.

Applicable to practical problems like peer-to-peer energy sharing.

Abstract

Finding a maximum-weight matching is a classical and well-studied problem in computer science, solvable in cubic time in general graphs. We consider the specialization called assignment problem where the input is a bipartite graph, and introduce in this work the ``discovery'' variant considering edge weights that are not provided as input but must be queried, requiring additional and costly computations. We develop here discovery algorithms aiming to minimize the number of queried weights while providing guarantees on the computed solution. We first show in this work the inherent challenges of designing discovery algorithms for general assignment problems. We then provide and analyze several efficient greedy algorithms that can make use of natural assumptions about the order in which the nodes are processed by the algorithms. Our motivations for exploring this problem stem from finding practical solutions to a variation of maximum-weight matching in bipartite hypergraphs, a problem recently emerging in the formation of peer-to-peer energy sharing communities.