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Automated negotiation has been used in a variety of distributed settings, such as privacy in the Internet of Things (IoT) devices and power distribution in Smart Grids. The most common protocol under which these agents negotiate is the Alternating Offers Protocol (AOP). Under this protocol, agents cannot express any additional information to each other besides a counter offer. This can lead to unnecessarily long negotiations when, for example, negotiations are impossible, risking to waste bandwidth that is a precious resource at the edge of the network. While alternative protocols exist which alleviate this problem, these solutions are too complex for low power devices, such as IoT sensors operating at the edge of the network. To improve this bottleneck, we introduce an extension to AOP called Alternating Constrained Offers Protocol (ACOP), in which agents can also express constraints to each other. This allows agents to both search the possibility space more efficiently and recognise impossible situations sooner. We empirically show that agents using ACOP can significantly reduce the number of messages a negotiation takes, independently of the strategy agents choose. In particular, we show our method significantly reduces the number of messages when an agreement is not possible. Furthermore, when an agreement is possible it reaches this agreement sooner with no negative effect on the utility.