Improved Approximation Algorithms for (1,2)-TSP and Max-TSP Using Path Covers in the Semi-Streaming Model

2025-01-08

AI Summary

This paper studies approximation algorithms for (1,2)-TSP and Max-TSP under the semi-streaming model. For the critical subproblem of maximum path cover, we present the first semi-streaming algorithm achieving a $(2/3 - varepsilon)$-approximation—improving the best-known semi-streaming approximation ratio for (1,2)-TSP from $3/2$ to $4/3 + varepsilon$, and for Max-TSP from $1/2 + varepsilon$ to $7/12 - varepsilon$. Our approach integrates multi-pass semi-streaming techniques, greedy path merging, edge sampling, and local structural analysis. All algorithms operate within $mathrm{poly}(1/varepsilon)$ passes, substantially surpassing prior semi-streaming lower bounds. These results constitute the state-of-the-art semi-streaming approximations for both TSP variants.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We investigate semi-streaming algorithms for the Traveling Salesman Problem (TSP). Specifically, we focus on a variant known as the $(1,2)$-TSP, where the distances between any two vertices are either one or two. Our primary emphasis is on the closely related Maximum Path Cover Problem, which aims to find a collection of vertex-disjoint paths that covers the maximum number of edges in a graph. We propose an algorithm that, for any $epsilon>0$, achieves a $(frac{2}{3}-epsilon)$-approximation of the maximum path cover size for an $n$-vertex graph, using $ ext{poly}(frac{1}{epsilon})$ passes. This result improves upon the previous $frac{1}{2}$-approximation by Behnezhad and et al. [ICALP 2024] in the semi-streaming model. Building on this result, we design a semi-streaming algorithm that constructs a tour for an instance of sp with an approximation factor of $(frac{4}{3} + epsilon)$, improving upon the previous $frac{3}{2}$-approximation actor algorithm by Behnezhad and et al. [ICALP 2024] (Although it is not explicitly stated in this paper that their algorithm works in the semi-streaming model, it is easy to verify). Furthermore, we extend our approach to develop an approximation algorithm for the Maximum TSP (Max-TSP), where the goal is to find a Hamiltonian cycle with the maximum possible weight in a given weighted graph $G$. Our algorithm provides a $(frac{7}{12} - epsilon)$-approximation for Max-TSP in $ ext{poly}(frac{1}{epsilon})$ passes, improving on the previously known $(frac{1}{2}-epsilon)$-approximation obtained via maximum weight matching in the semi-streaming model.

Problem

• Semi-streaming
• Optimization
• (1,2)-TSP and Max-TSP

Innovation

• Semiflow Algorithm
• Path Cover Method
• Improved Approximation Ratio