Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16893

Approximation Algorithms for Graphic Multi-Depot TSP and Graphic Multi-Path TSP

Niklas Dahlmeier·Raoul Luqué·Philipp Pabst·Laura Vargas Koch
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give improved approximation algorithms for two graphic TSP variants. First, we consider Multi-Depot TSP on graphic instances. In this variant, we are given an undirected graph with a set of depot vertices, and we are tasked with finding an Eulerian multi-edge set that connects each non-depot vertex to at least one depot. For this problem, Deppert, Kaul and Mnich (ESA 2023) presented a 3/2-approximation algorithm with running time exponential in the number of depots. We improve this result to a fully polynomial 3/2-approximation. Our algorithm is a variant of the classic Christofides-Serdyukov algorithm with an improved and simple analysis. On subcubic instances, we improve the approximation guarantee to 4/3 using the technique of removable pairings introduced by Mömke and Svensson (J. ACM 2016). We use a removable pairing to construct a set of disjoint cycles that can be completed to a cheap feasible solution. Second, we consider the Graphic Multi-Path TSP, which was previously considered by Alimi, Dahlmeier, Mömke, Pabst and Vargas Koch (SOSA 2026) who give a 2-approximation for this problem. Additionally, the authors present a reduction that shows that any better-than-two approximation for Graphic Multi-Depot TSP immediately leads to an improvement for Graphic Multi-Path TSP. We give an improved algorithm and analysis that yield an approximation guarantee of 1.9304.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Niklas Dahlmeier, Raoul Luqué, Philipp Pabst, Laura Vargas Koch. 2026-10-12. Approximation Algorithms for Graphic Multi-Depot TSP and Graphic Multi-Path TSP. https://arxiv.org/abs/2610.16893

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

O(1) Expected Insertion for Random Walk d-ary Cuckoo Hashing up to the Load Threshold

The random walk $d$-ary cuckoo hashing algorithm was defined by Fotakis, Pagh, Sanders, and Spirakis to generalize and improve upon the standard cuckoo hashing algorithm of Pagh and Rodler. Random walk $d$-ary cuckoo hashing has low space overhead, guaranteed fast access, and fast in practice insertion time. In this paper, we give a theoretical insertion time bound for this algorithm. More precisely, for every $d\ge 3$ random hashes, let $c_d^*$ be the sharp threshold for the load factor at which a valid assignment of $cm$ objects to a hash table of size $m$ exists with high probability. We show that for any $d\ge 3$ hashes and load factor $c<c_d^*$, the expectation of the random walk insertion time is $O(1)$, that is, a constant depending only on $d$ and $c$ but not $m$.

cs.DS↗
arXiv2026

Approximating Unsplittable Capacitated Vehicle Routing: Analysis via Factor-Revealing LPs

The capacitated vehicle routing problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. Given a depot and a set of customers with demands in a metric space, the objective is to find a set of tours, each starting and ending at the depot, operated by a capacitated vehicle located at the depot, such that all customers are served and the total travel cost is minimized. We consider the unsplittable variant, where the demand of each customer must be served entirely by a single tour. The previous best approximation ratio was about $α+1.694$, where $α$ is the approximation ratio for the traveling salesman problem. We propose improved approximation algorithms. First, we introduce a unified algorithmic framework that generalizes and integrates known linear programming (LP) rounding and matching-based approaches. We show that these techniques are complementary and can be combined to yield improved approximation ratios. Second, we develop a novel analytical framework based on customer demand discretization, which yields new structural bounds and, through factor-revealing LPs, achieves expected approximation ratios of about $α+1.647$ for the general-capacity case and $α+1.548$ for the fixed-capacity case. Finally, we derandomize our algorithms using a priori 2-tour sets and the method of conditional expectations while maintaining nearly the same approximation ratios.

cs.DS↗
arXiv2026

Entrywise Low-Rank Approximation and Matrix $p \rightarrow q$ Norms via Global Correlation Rounding

Given a matrix $A$, the goal of the entrywise low-rank approximation problem is to find $\operatorname{argmin} \|A-B\|_p$ over all rank-$k$ matrices $B$, where $\| \cdot \|_p$ is the entrywise $\ell_p$ norm. When $p = 2$ this well-studied problem is solved by the singular value decomposition, but for $p \neq 2$ the problem becomes computationally challenging. For every even $p > 2$ and every fixed $k$, we give the first polynomial-time approximation scheme for this problem, improving on the $(3 + \varepsilon)$ approximation of Ban, Bhattiprolu, Bringmann, Kolev, Lee, and Woodruff, the bi-criteria approximation of Woodruff and Yasuda, and the additive approximation scheme of Anderson, Bakshi, and Hopkins. Prior algorithmic approaches based on sketching and column selection, which yielded a polynomial-time approximation scheme in the $p < 2$ setting, face concrete barriers when $p > 2$. Instead, we use the Sherali-Adams hierarchy of convex programs, and in so doing suggest a methodology for how to use convex hierarchies to design polynomial-time approximation schemes for continuous optimization problems. We use the same algorithmic strategy to give a new family of additive approximation algorithms for matrix $p \rightarrow q$ norms, which are intimately related to small-set expansion and quantum information. In particular, we give the first nontrivial additive approximation algorithms in the regime $p < 2 < q$.

cs.DS↗