Selecting a Maximum Solow-Polasky Diversity Subset in General Metric Spaces Is NP-hard

Abstract: The Solow--Polasky diversity indicator (or magnitude) is a classical measure of diversity based on pairwise distances. It has applications in ecology, conservation planning, and, more recently, in algorithmic subset selection and diversity optimization. In this note, we investigate the computational complexity of selecting a subset of fixed cardinality from a finite set so as to maximize the Solow--Polasky diversity value. We prove that this problem is NP-hard in general metric spaces. The reduction is from the classical Independent Set problem and uses a simple metric construction containing only two non-zero distance values. Importantly, the hardness result holds for every fixed kernel parameter $θ_0>0$; equivalently, by rescaling the metric, one may fix the parameter to $1$ without loss of generality. A central point is that this is not a boilerplate reduction: because the Solow--Polasky objective is defined through matrix inversion, it is a nontrivial nonlinear function of the distances. Accordingly, the proof requires a dedicated strict-monotonicity argument for the specific family of distance matrices arising in the reduction; this strict monotonicity is established here for that family, but it is not assumed to hold in full generality. We also explain how the proof connects to continuity and monotonicity considerations for diversity indicators.

• The paper proves that the subset selection problem for maximizing Solow–Polasky diversity is NP-hard in general metric spaces.
• It employs a reduction from the Independent Set problem, translating graph adjacency into specific metric space distances to establish NP-hardness.
• The study highlights implications for diversity optimization, suggesting heuristic and approximation algorithms as practical alternatives.

NP-Hardness of Subset Selection for Solow–Polasky Diversity in General Metric Spaces

Background and Problem Formalization

The Solow–Polasky diversity indicator is a set-based diversity measure leveraging a parametrized exponential kernel of pairwise distances, widely adopted in ecology, conservation biology, and algorithmic diversity optimization. For a set $S = \{x_1, ..., x_k\}$ from a metric space $(X, d)$ and a fixed parameter $\theta > 0$, the Solow–Polasky diversity is quantified as $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$, where $Z_{ij} = e^{-\theta d(x_i, x_j)}$. Selection of a $k$-subset maximizing this diversity criterion is relevant in both multi-objective optimization and combinatorial subset selection contexts.

The key open question, explicitly posed by Pereverdieva et al., was whether the subset selection problem maximizing Solow–Polasky diversity is computationally hard in general metric spaces. Previous literature settled NP-hardness for other diversity measures such as Riesz $s$-energy, max–min, and max–sum objectives in similar settings, but the nonlinearity and nonlocality of the Solow–Polasky indicator (due to the matrix inversion step) left its complexity unresolved.

Formally, the decision version is: given $(X, d)$, $k$, $\theta_0 > 0$, and threshold $(X, d)$0, decide if there exists $(X, d)$1, $(X, d)$2, with $(X, d)$3.

Reduction and Core Technical Result

The NP-hardness is established via a reduction from the classical Independent Set problem. The reduction constructs a metric space over the vertex set $(X, d)$4 of a graph $(X, d)$5, encoding adjacency into the distances:

• $(X, d)$6 if $(X, d)$7,
• $(X, d)$8 if $(X, d)$9,
• $\theta > 0$0 if $\theta > 0$1, $\theta > 0$2,

where $\theta > 0$3 is chosen to produce similarity values $\theta > 0$4 and $\theta > 0$5, ensuring off-diagonal similarities are sufficiently small for the strict monotonicity argument.

An independent set $\theta > 0$6 of $\theta > 0$7 corresponds to a $\theta > 0$8-subset in $\theta > 0$9 with all pairwise distances $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$0, inducing a similarity matrix $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$1, for which $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$2. This value strictly dominates any $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$3-subset containing adjacent vertices, whose similarity matrices include larger off-diagonal entries $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$4.

Strict Monotonicity Analysis

A fundamental difficulty is that the Solow–Polasky objective depends nonlocally on the input metric, complicating standard monotonicity-based reductions. The analysis in the paper sidesteps the lack of global strict monotonicity by establishing this property for the structured family of similarity matrices arising from the reduction. By introducing a parameterized deformation (interpolating a distance from $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$5 to $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$6), the derivative of the objective, $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$7, is shown to be strictly positive. This ensures subsets containing edges are strictly suboptimal.

The analysis is aided by a careful bounding of matrix norms, guaranteeing invertibility and strict positivity of the relevant weight vector $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$8 via the Neumann series.

Context within Diversity Optimization

This result situates Solow–Polasky diversity subset selection parallel to max–sum, max–min, and Riesz energy–based diversity objectives, for which NP-hardness has already been established in the literature [Pereverdieva et al., 2025]. While max–sum and related objectives permit straightforward reductions via their direct dependence on pairwise distances, the Solow–Polasky formulation's reliance on the matrix inverse introduces nontrivial analytical complications, resolved here by ad hoc structural analysis restricted to the constructed instances.

Moreover, the result holds for all fixed values of the kernel parameter $SP_\theta(S) = \mathbf{1}^\top Z^{-1}\mathbf{1}$9, as the input metric can always be rescaled, eliminating parameter-specific loopholes.

Implications and Future Directions

The practical consequence is that exact maximization of Solow–Polasky diversity for cardinality-constrained subset selection is infeasible in general metric spaces unless P = NP. Heuristic and approximation algorithms, such as the greedy deletion strategies suggested by previous work, remain the only viable alternatives in high-cardinality or large-ground-set regimes.

On the theoretical side, this result closes an explicit complexity-theory gap for general-metric diversity optimization, thereby completing the NP-hardness landscape for most major set-based diversity indicators.

Natural extensions include investigation of approximation algorithms and inapproximability thresholds, parameterized complexity analyses (aligned with emerging FPT studies of solution diversity [Baste et al., 2022]), and the identification of tractable subclasses (such as fixed-dimensional Euclidean or doubling metrics). The nontrivial monotonicity behavior points to the need for fine-grained analysis of the indicator's properties, particularly for understanding algorithmic implications of its connection to the theory of metric space magnitude [Leinster, 2021; Huntsman, 2023].

Conclusion

Maximizing the Solow–Polasky diversity for cardinality-constrained subset selection is NP-hard in general metric spaces for any fixed kernel parameter. The reduction leverages non-orthodox monotonicity arguments and specialized structural properties of the indicator, reinforcing that the complexity of set-based diversity maximization is a robust phenomenon, largely immune to the choice of reasonable distance-based diversity metric. Prospects for future research include approximation schemes, fixed-parameter tractability, and extension to other kernel-based or matrix-function–based diversity measures.

For further details, see "Selecting a Maximum Solow-Polasky Diversity Subset in General Metric Spaces Is NP-hard" (2604.05495).