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 (32−ε)
Nearby in the stack
-approximation of the maximum path cover size for an
n
-vertex graph, using
poly(ε1)
passes. This result improves upon the previous
21
-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
(34+ε)
, improving upon the previous
23
-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
(127−ε)
-approximation for Max-TSP in
poly(ε1)
passes, improving on the previously known
(21−ε)
-approximation obtained via maximum weight matching in the semi-streaming model.