On the integrality gap of small Asymmetric Traveling Salesman Problems: A polyhedral and computational approach

Eleonora Vercesi, János Barta, Luca Maria Gambardella, Stefano Gualandi, Monaldo Mastrolilli

Abstract: In this paper, we investigate the integrality gap of the Asymmetric Traveling Salesman Problem (ATSP) with respect to the linear relaxation given by the Asymmetric Subtour Elimination Problem (ASEP) for instances with n<math><mi is="true">n</mi></math> nodes, where n<math><mi is="true">n</mi></math> is small. In particular, we focus on the geometric properties and symmetries of the ASEP polytope (PASEPn<math><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">ASEP</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msubsup></math>) and its vertices. The polytope’s symmetries are exploited to design a heuristic pivoting algorithm to search vertices where the integrality gap is maximized. Furthermore, a general procedure for the extension of vertices from PASEPn<math><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">ASEP</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msubsup></math> to PASEPn+1<math><msubsup is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">ASEP</mi></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">+</mo><mn is="true">1</mn></mrow></msubsup></math> is defined. The generated vertices improve the known lower bounds of the integrality gap for 16≤n≤22<math><mrow is="true"><mn is="true">16</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">≤</mo><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≤</mo><mn is="true">22</mn></mrow></math> and, provide small hard-to-solve ATSP instances.

External IDs:dblp:journals/disopt/VercesiBGGM25