Polynomial-Time Riesz-Energy Subset Selection for Ordered Point Sets on Lines and $\ell_1$-Staircases

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Chinese explanation / 中文解读

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Original abstract

We study the one-dimensional fixed-cardinality minimum Riesz $s$-energy subset problem with fixed exponent $s > 0$: given ordered real points $x_1 < x_2 < \cdots < x_n$, a positive parameter $s>0$, and a cardinality $k$, choose indices $1 \leq i_1 < \cdots < i_k \leq n$ minimizing $E_s(i_1,\ldots,i_k)=\sum_{1\leq p<q\leq k}(x_{i_q}-x_{i_p})^{-s}$. The paper proves a Monge property for the one-dimensional Riesz interaction. Encoding feasible subsets by increasing index vectors, this Monge inequality implies submodularity on a finite distributive lattice and gives polynomial-time solvability via submodular minimization over distributive lattices. The structural construction is valid for every real $s>0$; bit-complexity claims require the arithmetic assumptions stated in the complexity section. The same structure also yields an explicit minimum $S$--$T$ cut algorithm with $k(n-k)$ threshold variables and $O(k^2(n-k)^2)$ finite pairwise edges. The resulting graph has $N=k(n-k)$ nodes and $M=O(k^2(n-k)^2)$ arcs after an $O(k^2(n-k)^2)$ coefficient-construction step; an $O(NM)$ max-flow bound gives an $O(k^3(n-k)^3)$ min-cut step, while the conservative $O(N^2M)$ bound gives $O(k^4(n-k)^4)$. Due to isometry, the results apply directly to subset selection on $\ell_1$ staircases, such as choosing diverse and representative Pareto front or skyline approximations in two dimensions. An open-source Python implementation of the min-cut algorithm accompanies the reproducibility material.

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