{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:37:47Z","timestamp":1753889867816,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Analysis of Algorithms","license":[{"start":{"date-parts":[[2018,7,31]],"date-time":"2018-07-31T00:00:00Z","timestamp":1532995200000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,7,31]],"date-time":"2018-07-31T00:00:00Z","timestamp":1532995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,7,31]],"date-time":"2018-07-31T00:00:00Z","timestamp":1532995200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>In a barter exchange market, agents bring items and seek to exchange their items with one another. Agents may agree to a k-way exchange involving a cycle of k agents. A barter exchange market can be represented by a digraph where the vertices represent items and the edges out of a vertex indicate the items that an agent is willing to accept in exchange for that item. It is known that the problem of finding a set of vertex-disjoint cycles with the maximum total number of vertices (MAX-SIZE-EXCHANGE) can be solved in polynomial time. We consider a barter exchange where each agent may bring multiple items, and items of the same agent are represented by vertices with the same color. A set of cycles is said to be tropical if for every color there is a cycle that contains a vertex of that color. We show that the problem of determining whether there exists a tropical set of vertex-disjoint cycles in a digraph (TROPICAL-EXCHANGE) is NP-complete and APX-hard. This is equivalent to determining whether it is possible to arrange an exchange of items among agents such that every agent trades away at least one item. TROPICAL-MAX-SIZE-EXCHANGE is a similar problem, where the goal is to find a set of vertex-disjoint cycles that contains the maximum number of vertices and also contains all of the colors in the graph. We show that this problem is likewise NP-complete and APX-hard. For the restricted case where there are at most two vertices of each color (corresponding to a restriction that each agent may bring at most two items), both problems remain NP-hard but are in APX. Finally, we consider MAX-SIZE-TROPICAL-EXCHANGE, where the set of cycles must primarily include as many colors as possible and secondarily include as many vertices as possible. We show that this problem is NP-hard.<\/jats:p><jats:p>Comment: Published in Discrete Mathematics and Theoretical Computer Science<\/jats:p>","DOI":"10.23638\/dmtcs-20-2-1","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:41:54Z","timestamp":1743698514000},"source":"Crossref","is-referenced-by-count":0,"title":["Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph: Barter Exchange with Multiple Items Per Agent"],"prefix":"10.23638","volume":"vol. 20 no. 2","author":[{"given":"Timothy","family":"Highley","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hoang","family":"Le","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2018,7,31]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1610.05115v5","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1610.05115v5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:41:54Z","timestamp":1743698514000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/3186"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,31]]},"references-count":0,"journal-issue":{"issue":"Analysis of Algorithms","published-online":{"date-parts":[[2018,7,31]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-20-2-1","relation":{"has-preprint":[{"id-type":"arxiv","id":"1610.05115v3","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1610.05115","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1610.05115","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2018,7,31]]},"article-number":"3186"}}