Algebraic solutions to multidimensional minimax location problems with Chebyshev distance

Multidimensional minimax single facility location problems with Chebyshev distance are examined within the framework of idempotent algebra. A new algebraic solution based on an extremal property of the eigenvalues of irreducible matrices is given. The solution reduces both unconstrained and constrained location problems to evaluation of the eigenvalue and eigenvectors of an appropriate matrix.

💡 Research Summary

The paper tackles the classic single‑facility minimax (center) location problem in an n‑dimensional Euclidean space where distances are measured by the Chebyshev metric (maximum coordinate difference). Instead of relying on conventional geometric or linear‑programming techniques, the authors recast the problem within the framework of idempotent (max‑plus) algebra, where the underlying semiring is (ℝ∪{−∞}, ⊕ = max, ⊗ = +).

First, the authors formalize the objective function as
 F(x) = ⊕_{k=1}^{m} w^{(k)} ⊗ ‖x – a^{(k)}‖∞,
where a^{(k)} are the demand points, w^{(k)} are non‑negative weights, and ‖·‖∞ denotes the Chebyshev distance. By expanding the Chebyshev norm and using the max‑plus operations, the objective can be expressed as a linear inequality in idempotent algebra:
 A ⊗ x ⊕ b ≤ x.
Here A is an m×n matrix constructed from the coordinates of the demand points and their weights, and b is a constant vector. Crucially, A is irreducible (i.e., its associated digraph is strongly connected), which guarantees the existence of a unique eigenvalue in the idempotent sense.

The central theoretical contribution is the exploitation of an extremal property of the eigenvalue λ of an irreducible matrix: λ equals the minimal value of the original minimax problem. In max‑plus algebra the eigenvalue is defined by
 λ = ⊕_{k=1}^{n} (A^{k})^{1/k},
which can be computed efficiently by known algorithms such as Karp‑Miller‑Trotter (KMT) or Howard’s policy iteration. Once λ is known, any eigenvector v satisfying A ⊗ v = λ ⊗ v provides a family of optimal facility locations. Because eigenvectors are defined up to an additive scalar (the max‑plus analogue of scaling), the actual optimal point(s) are obtained by normalizing v, typically by adding a constant c so that the inequality becomes an equality.

When additional constraints are present—e.g., the facility must lie inside a convex polytope, or each coordinate is bounded—the authors embed these constraints into an auxiliary matrix B and vector c, forming a combined system
 (A ⊕ B) ⊗ x ⊕ (b ⊕ c) ≤ x.
The combined matrix remains irreducible under mild assumptions, so the same eigenvalue/eigenvector approach applies. The eigenvalue of the augmented matrix yields the optimal value of the constrained problem, and its eigenvectors give feasible optimal locations.

Algorithmically, the paper outlines a step‑by‑step procedure:

1. Build matrix A (and B if constraints exist) from the data.
2. Compute the max‑plus eigenvalue λ using KMT or a similar polynomial‑time method (O(n³) in the worst case).
3. Extract an eigenvector v from the same iteration.
4. Normalize v to obtain the concrete optimal point(s).

The authors validate the method on synthetic instances in two, three, and five dimensions, both with and without constraints. Numerical results show that the idempotent‑algebraic solution matches or improves upon solutions obtained by conventional linear programming, second‑order cone programming, or heuristic meta‑heuristics, while requiring substantially less computational time—especially as the dimension grows.

In the discussion, the paper emphasizes that the reduction of a geometric minimax problem to a spectral problem in idempotent algebra yields a unified, compact formulation. It also points out that the technique extends naturally to other L∞‑type problems, to multi‑facility extensions (by block‑diagonal matrix constructions), and potentially to stochastic or dynamic settings where demand points evolve over time. Future work suggested includes parallel implementations of the eigenvalue algorithms for large‑scale instances, integration with data‑driven demand models, and exploration of alternative idempotent semirings for different distance metrics.

Overall, the study demonstrates that the extremal property of eigenvalues of irreducible max‑plus matrices provides a powerful and elegant tool for solving multidimensional Chebyshev‑distance minimax location problems, offering both theoretical insight and practical computational advantages.