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In a barter exchange market, agents bring items and seek to exchange theiritems with one another. Agents may agree to a k-way exchange involving a cycleof k agents. A barter exchange market can be represented by a digraph where thevertices represent items and the edges out of a vertex indicate the items thatan agent is willing to accept in exchange for that item. It is known that theproblem of finding a set of vertex-disjoint cycles with the maximum totalnumber of vertices (MAX-SIZE-EXCHANGE) can be solved in polynomial time. Weconsider a barter exchange where each agent may bring multiple items, and itemsof the same agent are represented by vertices with the same color. A set ofcycles is said to be tropical if for every color there is a cycle that containsa vertex of that color. We show that the problem of determining whether thereexists a tropical set of vertex-disjoint cycles in a digraph(TROPICAL-EXCHANGE) is NP-complete and APX-hard. This is equivalent todetermining whether it is possible to arrange an exchange of items among agentssuch that every agent trades away at least one item. TROPICAL-MAX-SIZE-EXCHANGEis a similar problem, where the goal is to find a set of vertex-disjoint cyclesthat contains the maximum number of vertices and also contains all of thecolors in the graph. We show that this problem is likewise NP-complete andAPX-hard. For the restricted case where there are at most two vertices of eachcolor (corresponding to a restriction that each agent may bring at most twoitems), both problems remain NP-hard but are in APX. Finally, we considerMAX-SIZE-TROPICAL-EXCHANGE, where the set of cycles must primarily include asmany colors as possible and secondarily include as many vertices as possible.We show that this problem is NP-hard.

Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph: Barter Exchange with Multiple Items Per Agent