Minimum Riesz s-Energy Subset Selection in Ordered Point Sets via Dynamic Programming

TL;DR

This paper introduces a dynamic programming method for selecting representative subsets with minimal Riesz s-energy in ordered point sets, effective in one-dimensional and certain two-dimensional cases, with proven near-optimality and practical efficiency.

Contribution

The paper develops an efficient dynamic programming algorithm for subset selection minimizing Riesz s-energy, extending its application to biobjective optimization and Pareto front representations.

Findings

The algorithm runs in O(n^2 k) time.

It often yields near-optimal solutions in practice.

The approach avoids common greedy selection mistakes.

Abstract

We present a dynamic programming algorithm for selecting a representative subset of size $k$ from a given set with $n$ points such that the Riesz $s$-energy is near minimized. While NP-hard in general dimensions, the one-dimensional case can use the natural data ordering for efficient dynamic programming as an effective heuristic solution approach. This approach is then extended to problems related to two-dimensional Pareto front representations arising in biobjective optimization problems. Under the assumption of sorted (or non-dominated) input, the method typically yields near-optimal solutions in most cases. We also show that the approach avoids mistakes of greedy subset-selection by means of example. However, as we demonstrate, there are exceptions where DP does not identify the global minimum; for example, in one of our examples, the DP solution slightly deviates from the configuration found by a brute-force search. This is because the DP scheme's recurrence is approximate. The total time complexity of our algorithm is shown to be $O(n^2 k)$. We provide computational examples with discontinuous Pareto fronts and an open-source Python implementation, demonstrating the approximate DP algorithm's effectiveness across various problems with large point sets.