Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual Algorithms (1612.07925v2)

Abstract: Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best known algorithm for $k$-means with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of $k$-means and (2) to satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a $6.357$-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of $k$-means where the underlying metric is not required to be Euclidean and for $k$-median in Euclidean metrics.

Improved Approximation Algorithms for Clustering via Primal-Dual Techniques

The paper under review introduces a novel primal-dual algorithmic framework for solving the classic $k$-Means and Euclidean $k$-Median clustering problems. The authors propose a new class of approximation algorithms that significantly improve over previous bounds by exploiting the geometric structure of these problems, and notably, do so within a primal-dual setting that has been traditionally resistant to such improvements.

Novel Contributions and Results

The focal point of the research is a new primal-dual methodology that advances the approximation ratio for $k$-Means from the longstanding $9 + \epsilon$ to approximately $6.357$. This result emanates from a meticulous refinement of the primal-dual approach which exploits the geometric properties inherent in the $k$-Means problem. This improvement overcomes the traditional barriers associated with non-metric spaces by introducing more "aggressive" opening strategies for facilities through a parameter $\delta$. For the Euclidean $k$-Median problem, a $2.633$ approximation ratio is achieved, striking a balance between the use of geometric insights and the robust primal-dual framework commonly employed for other metric facility location problems.

Technical Approach

The paper significantly innovates by modifying the classical primal-dual schema, introducing a parameter $\delta$ which regulates facility activation — a move away from the algorithm's reliance on local search heuristics and previously used global LP-rounding techniques. This parameterized primal-dual approach allows the algorithm to take advantage of the Euclidean space properties—particularly the approximation efficiency gains obtainable through simpler facility-distance relationships, enabling a more granular control of the facility-contribution tradeoff.

Additionally, the authors present a quasi-polynomial algorithm for addressing the $k$-Means clustering problem, subsequently developing it into a fully polynomial-time approximation scheme (FPTAS) through strategic parameter tuning and dependency analysis. This FPTAS narrows the interval within which possible solutions are explored, ensuring computational feasibility without sacrificing approximation quality.

Implications and Future Directions

This work's implications extend to both theoretical and practical spheres. Theoretically, by enhancing our understanding of primal-dual methods in non-metric spaces, it opens avenues for applying similar strategies to other combinatorial optimization challenges that benefit from Euclidean properties. Practically, this translates to reduced computational complexity and better performance in machine learning applications where clustering is pivotal.

Looking forward, the results suggest several intriguing directions. One avenue is the exploration of whether similar primal-dual enhancements can be applied to other graph-theoretical problems outside the traditional field catered by the triangle inequality. Moreover, further tuning of the parameter $\delta$ and its relatives could yield improved algorithms for a broader class of clustering problems, potentially even outside Euclidean spaces.

The paper also prompts the exploration of how these primal-dual methods can embrace data-dependent strategies, adapting dynamically to the problem instance's structure in real-time settings, which is particularly pertinent for large-scale data science applications.

In summary, this paper makes a significant stride in approximating $k$-Means and $k$-Median problems, extending the utility of primal-dual techniques and setting a foundation for future enhancements and applications in clustering and beyond.