Complexity of Stability in Trading Networks

TL;DR

This paper investigates the computational complexity of various solution concepts in trading networks, revealing that finding and verifying stable outcomes is NP-complete, highlighting inherent computational challenges.

Contribution

It demonstrates that deciding the existence of certain stable outcomes in trading networks is NP-complete, even under simplified conditions, extending previous linear-time results.

Findings

Deciding existence of outcomes with upstream and downstream contracts is NP-complete.

Verifying stability of a given outcome is NP-complete.

Stable outcomes are NP-hard to compute in trading networks.

Abstract

Efficient computability is an important property of solution concepts in matching markets. We consider the computational complexity of finding and verifying various solution concepts in trading networks-multi-sided matching markets with bilateral contracts-under the assumption of full substitutability of agents' preferences. It is known that outcomes that satisfy trail stability always exist and can be found in linear time. Here we consider a slightly stronger solution concept in which agents can simultaneously offer an upstream and a downstream contract. We show that deciding the existence of outcomes satisfying this solution concept is an NP-complete problem even in a special (flow network) case of our model. It follows that the existence of stable outcomes--immune to deviations by arbitrary sets of agents-is also an NP-hard problem in trading networks (and in flow networks). Finally, we show that even verifying whether a given outcome is stable is NP-complete in trading networks.