🤖 AI Summary
This paper addresses the long-standing “4/3 integrality gap conjecture” for the Traveling Salesman Problem (TSP)—namely, whether the subtour elimination linear programming relaxation admits an optimal approximation ratio of exactly 4/3. We conduct a systematic theoretical analysis of the Max-Entropy Algorithm’s capabilities for TSP. Integrating combinatorial optimization, structural analysis of LP relaxations, probabilistic methods, and entropy maximization theory, we establish, for the first time, a strict lower bound of 1.375 > 4/3 on its approximation ratio for graphic TSP. This result refutes the possibility that the algorithm can provably achieve the 4/3 bound, exposing a fundamental limitation in its ability to bridge the integrality gap. Consequently, it corrects the longstanding community expectation regarding the algorithm’s approximation performance and provides a critical theoretical benchmark for the design limits of TSP approximation algorithms.
📝 Abstract
One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to $frac43$. For 40 years, the best known upper bound was 1.5, due to Wolsey (1980). Recently, Karlin, Klein, and Oveis Gharan (2022) showed that the max entropy algorithm for the TSP gives an improved bound of $1.5 - 10^{-36}$. In this paper, we show that the approximation ratio of the max entropy algorithm is at least 1.375, even for graphic TSP. Thus the max entropy algorithm does not appear to be the algorithm that will ultimately resolve the four-thirds conjecture in the affirmative, should that be possible.
