Cooperating to buy shoes: an application of picking cycles in directed graphs

• Authors: I. D. Sanders
• Venue: Proceedings of the South African Institute for Computer Scientists and Information Technologists Conference
• Year: 2013

In this paper cycle picking in defined as an optimisation problem where cycles are chosen from a directed graph under the constraint that any node that is in a cycle in the original directed graph must be in at least one of the chosen cycles. This problem is an abstraction of a real-world problem where people who require different sized shoes cooperate to be able to buy shoes that fit. The paper considers a number of variations of the problem -- the minimum number of cycles, the all small cycles and the minimum total cycle length views. The paper also shows how the directed graph can be amended to ensure that every node in the original graph is in at least one cycle of an augmented graph before the optimisation phase commences. This is the equivalent of adding dummy people to the group of all people who need shoes to ensure that everyone who needs shoes has a group (even if it contains dummy nodes) of other people with whom to cooperate. Finally a simple greedy heuristic approach for finding approximate solutions to instances of the problem is considered.