Source-linked AI summary
Exact algorithms for optimal discretization
László Kozma, Junqi Tan
TL;DR
The paper studies planar separation using axis-parallel lines, including optimal discretization motivated by machine-learning preprocessing and the related point-separation problem. Structural bounds on the smaller orientation-specific separator set reduce the search space, yielding improved exponential-time algorithms for both problems.
Problem
Optimal discretization uses horizontal and vertical thresholds to ensure differently labeled data points do not share a discretized cell, while point separation requires every induced cell to contain at most one point.
Method
The algorithms exploit structural bounds showing that an optimal solution has few lines in one orientation, thereby reducing the search space.
Results
The optimal discretization problem is solvable in time O(1.9602^n), while point separation is solvable in time O(1.8906^n) without assuming general position.
Takeaways & Limitations
The structural bounds provide improved exponential running times for both optimal discretization and point separation.
Takeaways & Limitations
The gap between 0.3n and 0.4n for the smaller separator-set cardinality in the two-colored case remains open.
Abstract
from arXiv · showhide
The optimal discretization problem asks, given two disjoint sets of points $R$ and $B$ in the plane, for a minimal family of horizontal and vertical lines that separate the two sets, so that no cell delimited by the lines contains points from both sets. The problem arises as a pre-processing in supervised machine learning, and has received significant attention in parameterized algorithmics. Answering the question raised by Bonnet, Giannopoulos, and Lampis [IPEC 2017] and Froese [PhD thesis, 2018], it was shown by Kratsch, Masařík, Muzi, Pilipczuk, and Sorge [SODA 2021] that optimal discretization admits a fixed-parameter algorithm with running time $2^{\mathcal{O}(k^2 \log k)} \cdot n^{\mathcal{O}(1)}$, where $k$ is the solution size and $n = |R| + |B|$. In this paper we give an algorithm for optimal discretization that runs in time $\mathcal{O}(1.9602^n)$. We also study the related point separation problem that asks to separate all input points by axis-parallel lines. For this problem we obtain an algorithm with runtime $\mathcal{O}(1.8906^n)$. Our guarantees follow from structural observations about bichromatic and monochromatic point sets, and hold even if points are allowed to share coordinates. To our knowledge, these are the first improvements over the trivial $2^n$ bound for both problems.
1 Introduction
This paper develops faster exact algorithms for optimal discretization and point separation by exploiting structural bounds on one orientation of the separator lines. The resulting algorithms improve the trivial 2^n-time barrier, including when points share coordinates.
- Problem and motivation: Optimal discretization separates two disjoint point sets using horizontal and vertical lines so each induced cell contains points from at most one set.Its original motivation is preprocessing continuous features into discrete values for supervised machine learning.
- Baseline and motivation: For optimal discretization, the baseline searches subsets of line locations, while guessing one orientation allows the other to be placed greedily in polynomial time.Trying the smaller orientation improves the baseline only to O*(n^k/2), which beats 2^n only by a polynomial factor.
- Results: O(1.9602^n) solves optimal discretization, while point separation takes O(1.8899^n) in general position and O(1.8906^n) otherwise.The guarantees also apply when input points share horizontal or vertical coordinates, with additional structural handling for forced lines.
- Problem and motivation: Point separation uses axis-parallel lines avoiding a single point set, requiring every induced cell to contain at most one point.It is also a multiclass version of discretization in which each point has a distinct class label.
- Significance: These are the first algorithms below the 2^n barrier for both problems, a central objective in exponential algorithm design.The paper places these results alongside earlier sub-2^n algorithms for several classical problems.
- Structural approach: The paper's central structural observation is that an optimal solution can have at most 0.4n lines in one orientation for optimal discretization and at most n/3 for point separation.These bounds reduce the search space; the point-separation bound is tight, while the two-colored bound lies between known 0.3n and 0.4n limits.
2 Algorithm for optimal discretization
The algorithm searches over one orientation of separators and greedily computes an optimal complementary set, then reduces the search space using balance bounds. These bounds yield an O(1.9602^n) running time and remain valid when input points share coordinates.
- Baseline algorithm: Given horizontal separators H, a greedy interval-hitting algorithm computes vertical separators V′ that are feasible and satisfy |V′| ≤ |V|.Colorful pairs not separated horizontally define x-coordinate intervals, and V′ hits all such intervals optimally.
- Baseline algorithm: The baseline enumerates up to n−1 separator locations for the smaller orientation and returns the feasible solution of minimum total size.It considers subsets of up to αn locations and uses the greedy complementary procedure for each choice.
- General position: The Balance Lemma guarantees an optimal solution in general position with min{|H|, |V|} ≤ 0.4n.Each vertical line uniquely separates at least one colorful pair, while each horizontal line uniquely separates at least two, enabling the balance argument.
- General position: Using α = 0.4, the binomial search sum gives a total running time of O(1.9602^n).The estimate uses H(0.4) ≤ 0.970952.
- General position: The 0.4n balance bound is close to optimal because an infinite family of general-position instances forces min{|H|, |V|} ≥ 0.3n.In the construction, every optimal solution has 3t horizontal and 3t vertical lines, giving min{|H|, |V|} = 0.3n.
3 Algorithm for point separation
The point separation algorithm reduces the search space by proving that an optimal solution has a sufficiently small horizontal or vertical side. This yields sub-2^n running times, including bounds that remain valid when coordinates are shared.
- Point separation asks for axis-parallel lines such that every pair of input points is separated by a horizontal or vertical line.
- The baseline algorithms run in O*(2^n) or O*(n^(k/2)), and the improved approach preserves their search structure while shrinking one guessed side.
- General position: For general-position instances, some optimal solution satisfies min{|H|, |V|} ≤ n/3, a bound used to derive the improved runtime.
- General position: The n/3 balance bound is tight: for every n = 3k + 1, an instance exists whose every optimal solution has min{|H|, |V|} ≥ k.
- Shared coordinates: When points may share coordinates, degeneracy d = min{d_x, d_y} gives min{|H|, |V|} ≤ n/3 + 2d/3 for some optimal solution.
4 Conclusions
The paper leaves structural and algorithmic gaps open, including the two-colored balance bound, the effect of general position, and possible further exponential-time improvements.
- The two-colored case retains a gap between 0.3n and 0.4n for the optimal smaller separator set, with n/3 identified as a plausible target.
- Point separation still has a gap between the bounds obtained with and without the general-position assumption.
- Further runtime improvements remain open, and a 2^o(n)-time algorithm has not been ruled out.
- Separating points with axis-parallel line segments forming a rectangulation likely requires different techniques and supports distinct minimization objectives.