Junqi Tan | Searcharxiv

arXiv2026

Exact algorithms for optimal discretization

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, Masa\v{r}\'ik, 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.

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arXiv2025

Faster exponential algorithms for cut problems via geometric data structures

For many hard computational problems, simple algorithms that run in time $2^n \cdot n^{O(1)}$ arise, say, from enumerating all subsets of a size-$n$ set. Finding (exponentially) faster algorithms is a natural goal that has driven much of the field of exact exponential algorithms (e.g., see Fomin and Kratsch, 2010). In this paper we obtain algorithms with running time $O(1.9999977^n)$ on input graphs with $n$ vertices, for the following well-studied problems: - $d$-Cut: find a proper cut in which no vertex has more than $d$ neighbors on the other side of the cut; - Internal Partition: find a proper cut in which every vertex has at least as many neighbors on its side of the cut as on the other side; and - ($\alpha,\beta$)-Domination: given intervals $\alpha,\beta \subseteq [0,n]$, find a subset $S$ of the vertices, so that for every vertex $v \in S$ the number of neighbors of $v$ in $S$ is from $\alpha$ and for every vertex $v \notin S$, the number of neighbors of $v$ in $S$ is from $\beta$. Our algorithms are exceedingly simple, combining the split and list technique (Horowitz and Sahni, 1974; Williams, 2005) with a tool from computational geometry: orthogonal range searching in the moderate dimensional regime (Chan, 2017). Our technique is applicable to the decision, optimization and counting versions of these problems and easily extends to various generalizations with more fine-grained, vertex-specific constraints, as well as to directed, balanced, and other variants. Algorithms with running times of the form $c^n$, for $c<2$, were known for the first problem only for constant $d$, and for the third problem for certain special cases of $\alpha$ and $\beta$; for the second problem we are not aware of such results.

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