Dual Charging for Half-Integral TSP

Nathan Klein; Mehrshad Taziki

Home/Archive/arXiv

arXiv · 2507.17999

Dual Charging for Half-Integral TSP

Nathan Klein·Mehrshad Taziki

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Nathan Klein, Mehrshad Taziki. 2025-07-24. Dual Charging for Half-Integral TSP. https://arxiv.org/abs/2507.17999

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

KEEP EXPLORING

Related papers

arXiv2026

An efficient implementation for solving the all pairs minimax path problem in an undirected dense graph

We provide an efficient $ O(n^2) $ implementation for solving the all pairs minimax path problem or widest path problem in an undirected dense graph. It is a code implementation of the Algorithm 4 (MMJ distance by Calculation and Copy) in a previous paper. The distance matrix is also called the all points path distance (APPD). We conducted experiments to test the implementation and algorithm, compared it with several other algorithms for solving the APPD matrix. Result shows Algorithm 4 works good for solving the widest path or minimax path APPD matrix. It can drastically improve the efficiency for computing the APPD matrix. There are several theoretical outcomes which claim the APPD matrix can be solved accurately in $ O(n^2) $ . However, they are impractical because there is no code implementation of these algorithms. It seems Algorithm 4 is the first algorithm that has an actual code implementation for solving the APPD matrix of minimax path or widest path problem in $ O(n^2) $, in an undirected dense graph.

cs.DS↗

arXiv2026

Approximating Partition in Deterministic Near-Linear Time

We propose a deterministic $\widetilde{O}(n + \frac{1}ε)$-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a polylogarithmic factor) assuming SETH (Strong Exponential Time Hypothesis) [Abboud, Bringmann, Hermelin, and Shabtay'22]. Prior to our work, the best known FPTAS for Partition runs in $\widetilde{O}(n + (\frac{1}ε)^{5/4})$ time [Deng, Jin and Mao'23, Wu and Chen'22]. Our result is obtained by solving a more general problem of weakly approximating Subset Sum.

cs.DS↗

arXiv2026

Dynamic Treewidth in Logarithmic Time

We present a dynamic data structure that maintains a tree decomposition of width at most $9k+8$ of a dynamic graph with treewidth at most $k$, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} \log n$, where $n$ is the number of vertices. The data structure also supports maintaining any ``dynamic programming scheme'' on the tree decomposition, providing, for example, a dynamic version of Courcelle's theorem with $O_{k}(\log n)$ amortized update time; the $O_{k}(\cdot)$ notation hides factors that depend on $k$. This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time $2^{k^{O(1)}} n^{o(1)}$. Furthermore, our data structure is arguably simpler. Our main novel idea is to maintain a tree decomposition that is ``downwards well-linked'', which allows us to implement local rotations and analysis similar to those for splay trees.

cs.DS↗