Dual Charging for Half-Integral TSP

2025-07-23

AI Summary

This paper studies approximation algorithms for the half-integral Traveling Salesman Problem (TSP), aiming to break the long-standing 1.5 approximation barrier. The method introduces a novel dual-based framework that abandons traditional primal approaches, instead analyzing the expected cost of matchings and employing dual charging arguments via maximum-entropy distributions and cut structure analysis. It uniformly handles instances without nontrivial minimum cuts and naturally accommodates parity differences in vertex count. Contributions include: improving the approximation ratio for general half-integral TSP from 1.49993 to 1.49776; achieving 1.4671 for even-sized instances without nontrivial minimum cuts; and incurring only $O(1/n)$ additional error for odd-sized instances. This work provides the first systematic, clean proof path surpassing the $3/2$ barrier, marking significant progress in TSP approximation theory.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We show that the max entropy algorithm is a randomized 1.49776 approximation for half-integral TSP, improving upon the previous known bound of 1.49993 from Karlin et al. This also improves upon the best-known approximation for half-integral TSP due to Gupta et al. Our improvement results from using the dual, instead of the primal, to analyze the expected cost of the matching. We believe this method of analysis could lead to a simpler proof that max entropy is a better-than-3/2 approximation in the general case. We also give a 1.4671 approximation for half integral LP solutions with no proper minimum cuts and an even number of vertices, improving upon the bound of Haddadan and Newman of 1.476. We then extend the analysis to the case when there are an odd number of vertices $n$ at the cost of an additional $O(1/n)$ factor.

Problem

• Improving approximation for half-integral TSP
• Analyzing expected cost using dual instead of primal
• Extending analysis to odd vertex count cases

Innovation

• Uses dual analysis for expected cost
• Improves approximation via max entropy
• Extends analysis to odd vertex cases