Finding the Best 3-OPT Move in Subcubic Time

• Main Authors: Giuseppe Lancia, Marcello Dalpasso
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Algorithms
• Subjects: 3-OPT; traveling salesman problem; local search
• Online Access: https://www.mdpi.com/1999-4893/13/11/306
• ISSN: 1999-4893

Given a Traveling Salesman Problem solution, the best 3-OPT move requires us to remove three edges and replace them with three new ones so as to shorten the tour as much as possible. No worst-case algorithm better than the <inline-formula><math display="inline"><semantics><mrow><mi>Θ</mi><mo>(</mo><msup><mi>n</mi><mn>3</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> enumeration of all triples is likely to exist for this problem, but algorithms with average case <inline-formula><math display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>n</mi><mrow><mn>3</mn><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> are not ruled out. In this paper we describe a strategy for 3-OPT optimization which can find the best move by looking only at a fraction of all possible moves. We extend our approach also to some other types of cubic moves, such as some special 6-OPT and 5-OPT moves. Empirical evidence shows that our algorithm runs in average subcubic time (upper bounded by <inline-formula><math display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>n</mi><mrow><mn>2</mn><mo>.</mo><mn>5</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>) on a wide class of random graphs as well as Traveling Salesman Problem Library (TSPLIB) instances.