arXiv subjects

Kangke Cheng

Publications and source records attributed to Kangke Cheng.

3 recordsLinked to original sources

arXiv2026

A Sub-4 Approximation for Fair $k$-Means

Fairness in clustering has attracted sustained research interest, motivated by the need to ensure equitable representation of protected groups in machine learning applications. We study fair $k$-means clustering in Euclidean space, where the proportion of each protected group in every cluster must lie within specified lower and upper bounds. These constraints make it challenging to determine both cluster centers and point assignments. We propose an approximation algorithm that combines a linear programming relaxation with geometric transformations of the input to construct candidate center sets. Given a $\rho$-approximate algorithm for weighted $k$-means and any $\epsilon>0$, our algorithm returns a fractional solution whose cost is at most $1+(3-1/\Gamma)\rho+O(\epsilon)$ times the optimal integral fair cost, where $\Gamma\approx6.357$ is an upper bound on the integrality gap of the standard Euclidean $k$-means LP. With a PTAS as the subroutine, the approximation ratio becomes $3.8427+O(\epsilon)$, improving the previous factor of $5+O(\epsilon)$ to below $4$. The solution satisfies all fairness constraints exactly and can be rounded to an integral assignment with a bounded additive violation of fairness and no increase in cost. The same approximation guarantee extends to the $k$-sparse Wasserstein barycenter problem.

cs.CG↗

arXiv2026

The Stretch Factor of Planar Delaunay Triangulations Is Less Than 1.65

Delaunay triangulations are a fundamental class of plane spanners, and determining their worst-case stretch factor has been a longstanding problem in computational geometry. We prove an upper bound of $1.65$, improving the bound of $1.998$ due to Xia (2011) and reducing the gap to the known lower bound of $1.5932$ by a factor of more than seven. Our proof works with the chains of circumdisks introduced by Xia, along which a path between two sites is assembled disk by disk. Xia measures such a path against a quantity attached to the whole chain, and because that quantity is not additive, his induction has to be carried alongside a separate global estimate. Our main idea is to measure the path against the progress it makes along the segment joining the two sites. This quantity is additive, so the bound becomes a Bellman recursion that forgets all but one number about the disks already passed, and we show that the bound holds if and only if a potential on the current state satisfies three local inequalities. The smallest feasible potential is the value function of that recursion, so searching for a potential becomes the problem of fitting this value function from above. The geometry of the disks reduces the fit to a linear program over functions of one variable, in which a GPT-based multi-agent system that we developed found a feasible point, certified in exact arithmetic.

cs.CG↗

arXiv2026

Sample-and-Search: An Effective Algorithm for Learning-Augmented k-Median Clustering in High dimensions

In this paper, we investigate the learning-augmented $k$-median clustering problem, which aims to improve the performance of traditional clustering algorithms by preprocessing the point set with a predictor of error rate $\alpha \in [0,1)$. This preprocessing step assigns potential labels to the points before clustering. We introduce an algorithm for this problem based on a simple yet effective sampling method, which substantially improves upon the time complexities of existing algorithms. Moreover, we mitigate their exponential dependency on the dimensionality of the Euclidean space. Lastly, we conduct experiments to compare our method with several state-of-the-art learning-augmented $k$-median clustering methods. The experimental results suggest that our proposed approach can significantly reduce the computational complexity in practice, while achieving a lower clustering cost.

cs.DS↗