Maximum Entropy is a 10/7-Approximation Algorithm for the TSP on Half-Integral Cycle Cut Instances

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 $\frac{4}{3}$. For 40 years, the best known upper bound was $1.5$, due to Wolsey. Recently, Karlin, Klein, and Oveis Gharan 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 maximum entropy algorithm is a $\frac{10}{7}$-approximation for half-integral cycle cut instances of the TSP. This class of instances contains examples which demonstrate the subtour LP has an integrality gap of at least $\frac{4}{3}$, as well as examples showing that the performance of the max entropy algorithm is no better than $\frac{11}{8}$. We note that in the authors recently gave an algorithm upper bounding the integrality gap of this class of instances by $\frac{4}{3}$, so this work does not (and could not) provide an improved bound on the integrality gap. However, since there is no reason to believe that the analysis of the maximum entropy algorithm on general instances is tight, our work provides hope (and potentially direction) for improved analysis on other instance classes.

• The paper proves that the max-entropy algorithm attains a 10/7-approximation on half-integral cycle cut TSP instances.
• It employs combinatorial decompositions and recursive probabilistic edge additions to achieve precise cost bounds.
• The analysis highlights a novel construction of stationary distributions, deepening insights into the integrality gap.

Maximum Entropy as a $\boldsymbol{10/7}$-Approximation Algorithm for TSP on Half-Integral Cycle Cut Instances

Overview

This work establishes that the maximum entropy (max-entropy) algorithm for the symmetric metric Traveling Salesman Problem (TSP) is a $\frac{10}{7}$-approximation algorithm on the class of half-integral cycle cut instances. The analysis provides a precise upper bound on the expected cost achieved by this randomized algorithm, strengthening the understanding of how max-entropy performs on key instance classes characterized by half-integrality and combinatorial structure. The results leverage combinatorial decompositions, probabilistic edge additions, and a detailed characterization of stationary distributions.

Context and Motivation

The TSP remains a central problem in combinatorial optimization, with LP relaxations playing a pivotal role in algorithm analysis. The integrality gap of the classic Subtour LP is conjectured to be $\frac{4}{3}$ (the four-thirds conjecture). The long-standing best ratio achieved by an efficient algorithm is $\frac{3}{2}$, due to Christofides [Christofides76], and this aligns with the worst-case integrality gap proved by Wolsey [Wolsey80]. Recent work marginally improved upon the $\frac{3}{2}$ factor by analyzing the max-entropy algorithm [KKO21], but such improvements on general instances remain microscopic.

Half-integral cycle cut instances are of particular interest: not only do they exhibit integrality gaps of at least $\frac{4}{3}$, but structural conjectures posit them as worst-case instances for the Subtour LP [SchalekampWvZ14]. Improving bounds specifically for these instances thus delivers meaningful theoretical progress with implications for the broader integrality gap conjecture.

Problem Definition and Algorithmic Setting

The TSP is defined on complete, edge-weighted graphs with the triangle inequality (metrics), and attention here is restricted to the symmetric case. Given the Subtour LP relaxation, a solution $x$ is half-integral if $x_e \in \{0, \frac{1}{2}, 1\}$ for all edges, and a cycle cut instance if every tight cut (for which $x(\delta(S))=2$, $|S|\ge2$) admits a partition into tight sets, recursively forming a hierarchical decomposition. These instances permit a clean combinatorial framework amenable to analysis.

The max-entropy algorithm samples from a maximum entropy distribution over spanning 1-trees (a spanning tree plus one extra edge, adhering to marginal constraints) and augments the resulting structure by adding a minimum-cost perfect matching on its set of odd-degree vertices. Prior results showed this algorithm attains slightly better than the $\frac{10}{7}$0-approximation for general metric TSP [KKO21].

Main Contributions

The paper’s principal theorem proves that on half-integral cycle cut instances, the max-entropy algorithm achieves an approximation factor of exactly $\frac{10}{7}$1. Moreover, there exist instances within this class for which the performance ratio of the algorithm is at least $\frac{10}{7}$2 (from previous work [JinKW25b]), showing that further improvement would require fundamentally different algorithmic tools or analytic insights.

Structural Decomposition and Sampling

A key technical step is transforming any half-integral cycle cut solution into a 4-regular multigraph by splitting $\frac{10}{7}$3 edges. The algorithm’s probabilistic structure is then described recursively over a binary hierarchy of tight cuts. At every level, choices regarding which edge(s) to add are made independently—this property is critical for the tractability and precision of the probabilistic analysis.

Stationary Distribution Construction

A novel aspect of the analysis is the explicit construction of stationary distributions over the multi-set of edges, maintained at each level of the cycle cut hierarchy. These distributions are engineered to guarantee:

• Every tight cut has either $\frac{10}{7}$4 or $\frac{10}{7}$5 edges of odd parity, with probabilities $\frac{10}{7}$6, $\frac{10}{7}$7, and $\frac{10}{7}$8, respectively.
• Each edge is included (counting possible doubles) in expectation exactly $\frac{10}{7}$9 times.
• The support is always on connected Eulerian multi-subgraphs.

The recursive extension ensures that sampled subgraphs can be completed to Eulerian tours via the addition of a minimum-cost perfect matching, and the expected cost is tightly bounded by $\frac{4}{3}$0 times the LP optimum.

Implications for Integrality Gap and Algorithmic Limits

Strikingly, the $\frac{4}{3}$1 ratio significantly improves the analysis for max-entropy on this instance class—while not improving the integrality gap (which is already known to be at most $\frac{4}{3}$2 on these instances [JinKW25]), it establishes that max-entropy’s performance far exceeds the general lower bound ($\frac{4}{3}$3 achievable) but does not reach the integrality gap itself. This clarification sets a new benchmark for probabilistic rounding algorithms on these structured instances.

Theoretical and Practical Implications

The findings delineate the precise capabilities and limitations of the maximum entropy algorithm for a major subclass of TSP instances, informing both analysis and algorithm design. Practically, this means that for TSP instances with optimal half-integral cycle cut LP solutions, max-entropy can be expected to deliver tours whose expected cost is no more than $\frac{4}{3}$4 times optimal. This is the first result to establish a constant below $\frac{4}{3}$5 for a large and structured class of instances unamenable to simpler analysis.

Theoretically, the work illuminates the role of structural decompositions and probabilistic independence in combinatorial optimization. The explicit recursive stationary distribution suggests new directions for extending similar analyses to broader or more general TSP instances. Additionally, the gap between the lower bound ($\frac{4}{3}$6) and the achieved factor ($\frac{4}{3}$7) raises precise questions about the tightness of this algorithmic paradigm and the fundamental approximability limits imposed by the combinatorial geometry of these LP relaxations.

Future Directions

• Tightening the analysis of the max-entropy rounding for other key special cases or for the general metric TSP, possibly by refining the coupling or stationary extension schemes.
• Understanding whether alternative structural decompositions or richer stationary distributions can bridge the gap to the conjectured $\frac{4}{3}$8 threshold.
• Exploring the application of these ideas to related combinatorial problems with similar LP hierarchies or decompositions, such as edge-cover and network design problems.
• Investigating whether the independence properties and local parity constraints leveraged here can be exploited algorithmically for heuristic or practical improvements.

Conclusion

This work substantiates that the max-entropy algorithm is a $\frac{4}{3}$9-approximation on half-integral cycle cut TSP instances, leveraging combinatorial hierarchy, recursive probabilistic enforcement of parity, and stationary distributions. The results reinforce the suitability of max-entropy-based approaches for structured relaxations, refine the limitations of current analysis, and suggest avenues for further improvements towards resolving the four-thirds conjecture and beyond.

References:

• "Maximum Entropy is a 10/7-Approximation Algorithm for the TSP on Half-Integral Cycle Cut Instances" (2607.01536)