{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:24:58Z","timestamp":1787340298583,"version":"build-2736575974"},"reference-count":16,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>We consider the gap between the cost of an optimal assignment in a complete bipartite graph with random edge weights, and the cost of an optimal traveling salesman tour in a complete directed graph with the same edge weights. Using an improved \u201cpatching\u201d heuristic, we show that with high probability the gap is $O((\\ln n)^2\/n)$, and that its expectation is $\\Omega(1\/n)$. One of the underpinnings of this result is that the largest edge weight in an optimal assignment has expectation $\\Theta(\\ln n \/ n)$. A consequence of the small assignment\u2013TSP gap is an $e^{\\tilde{O}(\\sqrt{n})}$\u2010time algorithm which, with high probability, exactly solves a random asymmetric traveling salesman instance. In addition to the assignment\u2013TSP gap, we also consider the expected gap between the optimal and second\u2010best assignments; it is at least $\\Omega(1\/n^2)$ and at most $O(\\ln n\/n^2)$.<\/jats:p>","DOI":"10.1137\/s0097539701391518","type":"journal-article","created":{"date-parts":[[2007,2,21]],"date-time":"2007-02-21T16:07:26Z","timestamp":1172074046000},"page":"1435-1452","source":"Crossref","is-referenced-by-count":9,"title":["The Probabilistic Relationship Between the Assignment and Asymmetric Traveling Salesman Problems"],"prefix":"10.1137","volume":"36","author":[{"given":"Alan","family":"Frieze","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gregory B.","family":"Sorkin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,1,26]]},"reference":[{"key":"R1","unstructured":"R. K. Ahuja, T. L. Magnanti, and J. B. Orlin,\n                      Network Flows Theory, Algorithms and Applications\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1993."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01192719"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177005647"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176989707"},{"key":"R5","unstructured":"E. Balas and P. Toth,\n                      Branch and bound methods\n                      , in The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, E. L. Lawler, J. K. Lenstra, A. H. Rinnooy Kan, and D. B. Shmoys, eds., Wiley, New York, 1986, pp. 361\u2013400."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.1019"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585751"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01589437"},{"key":"R9","first-page":"455","volume":"8","author":"Erdo\u02dds P.","year":"1964","journal-title":"Publ. Math. Inst. Hungar. Acad. Sci."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(85)90059-9"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539792235384"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0110015"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"R. M. Karp,\n                      An upper bound on the expected cost of an optimal assignment\n                      , in Discrete Algorithms and Complexity: Proceedings of the Japan\u2013U.S. Joint Seminar, D. Johnson et al., eds., Academic Press, New York, 1987, pp. 1\u20134.","DOI":"10.1016\/B978-0-12-386870-1.50006-X"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0208045"},{"key":"R15","unstructured":"R. M. Karp and J. M. Steele,\n                      Probabilistic analysis of heuristics\n                      , in The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, eds., Wiley, New York, 1985, pp. 181\u2013206."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02699376"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539701391518","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:55Z","timestamp":1787336935000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539701391518"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":16,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/S0097539701391518"],"URL":"https:\/\/doi.org\/10.1137\/s0097539701391518","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}