Qualitative Properties of k-Center Problems

This paper delves into the qualitative properties of generalized k-center problems, a fundamental class of facility location problems. Traditionally, the k-center problem aims to find k locations (centers) for facilities such that the maximum distance from any demand point to its nearest facility is minimized, often using Euclidean distances (balls). This work significantly extends this classic formulation by employing the Minkowski gauge function to define “generalized balls.” The Minkowski gauge p_F(x) for a set F measures how much F must be scaled to contain x. This allows for a much broader class of shapes for the “balls” (any compact convex set F containing the origin in its interior), moving beyond just standard Euclidean spheres.

The problem, even in its standard form, is known to be non-smooth and non-convex, making its analysis and numerical solution challenging. A key insight is that the objective function of this generalized k-center problem can be formulated within the framework of Difference-of-Convex (DC) programming, where the objective is expressed as a difference of two convex functions. This structural property is crucial for theoretical analysis.

The paper’s significance lies in providing a rigorous theoretical foundation for understanding the behavior of both global and local optimal solutions to this generalized k-center problem, rather than focusing on specific algorithms.

Key Innovations and Contributions:

1. Generalization of the k-Center Problem: The primary innovation is the systematic study of the k-center problem using Minkowski gauge functions. This broadens the applicability and theoretical scope beyond Euclidean metrics, enabling analysis with various non-Euclidean “distances” or facility coverage shapes.

2. Existence and Properties of Global Optimal Solutions:

   • The paper establishes the fundamental result that the set of global optimal solutions is non-empty and closed for the generalized k-center problem. This provides assurance that solutions exist and are well-behaved topologically.
   • For the one-center problem (k=1), a stronger result is shown: the global optimal solution set is compact and convex. This makes the k=1 case considerably more tractable.
   • Crucially, the paper demonstrates that for k > 1, the global optimal solution set is not necessarily compact (it can be unbounded), as illustrated by a concrete example. This highlights a critical difference in qualitative behavior as k increases.
   • A significant contribution is the characterization of compactness for k > 2: the global optimal solution set is compact if and only if the optimal value strictly decreases when an additional (k+1)-th center is allowed. This provides a theoretical condition for the well-behavedness of the solution set.
   • Even when compact (e.g., for d=1, k=2), the solution set can be disconnected and non-convex, underscoring the inherent complexity of the problem.

3. Existence and Characterization of Local Optimal Solutions:

   • The paper clearly defines local optimal solutions for the generalized k-center problem, demonstrating through examples that local optima are not necessarily global optima when k > 1.
   • A sufficient condition for a system of centers to be a local optimal solution is established. This condition is related to the uniqueness of “attractive” centers for each demand point and the property that each center locally optimizes its associated one-center problem. This provides a pathway to understanding and potentially identifying local optima.

Main Prior Ingredients Needed:

1. Classical k-Center Problem and Smallest Enclosing Circle Problem: The paper builds directly upon the established foundations of these problems, including their historical context (e.g., Sylvester’s problem). The challenges inherent in their non-smooth and non-convex nature are the starting point for this deeper qualitative analysis.

2. Minkowski Gauge Functions and Convex Analysis: A strong background in convex analysis is indispensable. The Minkowski gauge function is a central concept, and its properties (e.g., positive homogeneity, subadditivity, relationship to norms) are heavily leveraged. Concepts like convex sets, compact sets, separation theorems, and properties of convex functions are implicitly or explicitly used throughout the proofs.

3. Difference-of-Convex (DC) Programming: The recognition that the k-center objective function is a DC function is a critical theoretical tool. Properties of DC functions, such as the fact that the maximum of finitely many DC functions is DC, allow for structural analysis that would be much harder otherwise.

4. Variational Analysis and Nonsmooth Optimization: Given the non-smooth and non-convex nature of the problem, concepts from variational analysis (e.g., subgradients, properties of non-smooth functions) are foundational for analyzing optimality conditions and solution set properties.

5. Topological Concepts: Familiarity with topological notions like closedness, compactness, connectedness, and the Weierstrass theorem (for existence of minimizers on compact sets) is crucial for the qualitative analysis of solution sets.

By shedding light on these fundamental qualitative properties, this paper provides a valuable theoretical framework for the k-center problem, which can inform the development of more effective and robust algorithms in the future.