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

TL;DR

The paper studies semi-streaming algorithms for Maximum Path Cover (MPC) and its applications to the (1,2)-TSP and Max-TSP. It introduces a simple two-phase, contraction-based MPC algorithm that achieves a $\frac{2}{3}-\epsilon$-approximation in poly$(\frac{1}{\epsilon})$ passes, enabling a semi-streaming $\left(\frac{4}{3}+\epsilon\right)$-approximation for $(1,2)$-TSP and a $\left(\frac{7}{12}-\epsilon\right)$-approximation for Max-TSP. These results improve over prior $(\tfrac{1}{2})$- and $(\tfrac{3}{2})$-approximations and provide the first non-trivial semi-streaming Max-TSP guarantees. The work combines contraction-based MPC analysis with streaming maximum/matching subroutines to achieve multi-pass, low-space algorithms with practical impact for large-scale graph optimization tasks.

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 cover the maximum number of edges in a graph. We propose an algorithm that, for any $ε> 0$, achieves a $(\frac{2}{3}-ε)$-approximation of the maximum path cover size for an $n$-vertex graph, using $\text{poly}(\frac{1}ε)$ passes. This result improves upon the previous $\frac{1}{2}$-approximation by Behnezhad 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 $(1,2)$-TSP with an approximation factor of $(\frac{4}{3} + ε)$, improving upon the previous $\frac{3}{2}$-approximation actor algorithm by Behnezhad et al. [ICALP 2024] (Although it is not explicitly stated in the 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 $(\frac{7}{12} - ε)$-approximation for Max-TSP in $\text{poly}(\frac{1}ε)$ passes, improving on the previously known $(\frac{1}{2}-ε)$-approximation obtained via maximum weight matching in the semi-streaming model.

Figures (5)

• Figure 1: Different possible cases for $u$ and $v$.
• Figure 2: An example of a graph $\tilde{G}$ for which \ref{['alg1']} produces a path cover whose size is $\frac{2}{3}$ times the size of the MPC.
• Figure 3: $C^*$ remains a path cover after contraction on $\textcolor{rgb(255,0,0)}{C^* \cap M}$ (red edges).
• Figure 4: An example of $C^*$ and $M$ illustrating the steps in the proof of \ref{['approx-1/2+(1-s)/4']}.
• Figure 5: An example of a graph where \ref{['alg-repeat']} terminates after two iterations. The algorithm produces a $\dfrac{3}{4}$-approximation of the Maximum Path Cover (MPC).

Theorems & Definitions (1)

• proof