Lift-and-Project Integrality Gaps for the Traveling Salesperson Problem

We study the lift-and-project procedures of Lov{'a}sz-Schrijver and Sherali-Adams applied to the standard linear programming relaxation of the traveling salesperson problem with triangle inequality. For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the integrality gap of the standard relaxation is at least 2. We prove that after one round of the Lov{'a}sz-Schrijver or Sherali-Adams procedures, the integrality gap of the asymmetric TSP tour problem is at least 3/2, with a small caveat on which version of the standard relaxation is used. For the symmetric TSP tour problem, the integrality gap of the standard relaxation is known to be at least 4/3, and Cheung (SIOPT 2005) proved that it remains at least 4/3 after $o(n)$ rounds of the Lov{'a}sz-Schrijver procedure, where $n$ is the number of nodes. For the symmetric TSP path problem, the integrality gap of the standard relaxation is known to be at least 3/2, and we prove that it remains at least 3/2 after $o(n)$ rounds of the Lov{'a}sz-Schrijver procedure, by a simple reduction to Cheung’s result.

💡 Research Summary

The paper investigates how the Lovász‑Schrijver (LS) and Sherali‑Adams (SA) lift‑and‑project hierarchies affect the integrality gaps of the classic linear programming (LP) relaxation for the Traveling Salesperson Problem (TSP) under the triangle inequality. Four variants are considered: symmetric tour, symmetric path, asymmetric tour, and asymmetric path. For the symmetric tour, the well‑known lower bound of 4/3 on the LP gap (proved by Cheung) persists after o(n) rounds of LS, confirming that many rounds of the hierarchy do not close the gap. The authors extend this line of work to the symmetric path problem, showing via a simple reduction to Cheung’s construction that the 3/2 gap also survives o(n) LS rounds.

The main technical contribution concerns the asymmetric tour version. While the basic LP relaxation has an integrality gap of at least 2 (Charikar, Goemans, Karloff), the authors prove that after a single round of LS (equivalently one round of SA) the gap remains at least 3/2, provided one works with the weaker formulation of the standard relaxation (the version that drops one of the two degree‑balance constraints). To achieve this, they introduce a novel “frame” structure in directed graphs and use it to construct protection matrices required by the LS operator. These frames consist of carefully chosen collections of directed paths that satisfy the cone constraints of the lifted space while preserving the necessary symmetry and diagonal conditions. The resulting protection matrix forces any feasible solution of the lifted polytope to retain a value that is at least 3/2 times the optimal integer tour length.

The paper also discusses why the stronger version of the relaxation (with both degree constraints) appears to become weaker after lifting, and therefore the lower bound cannot be proved for that formulation with the current techniques.

Overall, the work demonstrates that even a single round of lift‑and‑project does not substantially improve the LP relaxation for asymmetric TSP, and that for symmetric variants the gap remains stubbornly large after many rounds. The frame‑based construction enriches the toolbox for designing protection matrices and may find applications in other combinatorial problems where directionality poses challenges. The results provide unconditional evidence that LP‑based rounding algorithms cannot break the 3/2 (symmetric path) or 3/2 (asymmetric tour) barriers without fundamentally new ideas.