A Lower Bound for the Max Entropy Algorithm for TSP

TL;DR

The paper studies the max entropy distribution used in rounding the Subtour LP for the TSP and proves a concrete lower bound: on graphic $k$-donut instances, the algorithm achieves an asymptotic ratio of $\frac{11}{8}=1.375$ rather than $\frac{4}{3}$, even when shortcutting is considered. It formalizes the graphic $k$-donut construction, analyzes the induced 1-tree and the minimum-cost $T$-join, and shows that the resulting Eulerian subgraph cannot be shortcut to a tour cheaper than $1.375$ times the optimum. By establishing both upper and lower asymptotic bounds and constructing adversarial shortcutting scenarios for two natural matchings $M_1$ and $M_2$, the work demonstrates that the max entropy approach is unlikely to resolve the four-thirds conjecture in the affirmative. The results highlight limitations of the current max entropy framework and motivate exploring algorithmic refinements or alternative rounding strategies to approach the conjectured bound.

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.

Figures (11)

• Figure 1: Illustration of the worst example known for the integrality gap for the symmetric TSP with triangle inequality. The figure on the left gives a graph, and costs $c_{ij}$ are the shortest path distances in the graph. The figure in the center gives the LP solution, in which the dotted edges have value 1/2, and the solid edges have value 1. The figure on the right gives the optimal tour. The ratio of the cost of the optimal tour to the value of the LP solution tends to 4/3 as $k$ increases.
• Figure 2: The $k$-donut instance.
• Figure 3: The max entropy algorithm begins by putting all of the 1-edges into the 1-tree. Then, one edge among the pair of dotted edges inside each red circled cut will be chosen independently. Next, one edge among each pair of dotted edges in the cycle resulting from contracting the red sets will be chosen independently. Finally, one of the two dotted edges incident to $w_0$ will be chosen independently, and the same for $w_1$.
• Figure 4: The edges of the $k$-donut incident to $u_0, u_1, \ldots, u_{2k-1}$.
• Figure 5: On the left is a 0-block, on the right is a 1-block. Note that $i$ is odd.

Theorems & Definitions (20)

• Theorem 1
• Definition 2: $k$-Donut Graph
• Definition 3: $k$-Donut Extreme Point
• Proposition 3
• Claim 4
• proof
• Claim 5
• proof
• Lemma 6
• proof