Computing Diverse and Nice Triangulations

2025-06-02

AI Summary

This paper studies the Diverse and High-Quality Triangulation (DNT) problem for simple $n$-gons: given a simple $n$-gon $P$, an integer $k geq 2$, a quality measure $sigma$, and an approximation factor $alpha geq 1$, compute $k$ pairwise distinct triangulations $T_1,dots,T_k$ such that $sigma(T_i) leq alpha sigma^*$ for all $i$ (where $sigma^*$ is the optimal value), while maximizing the sum of pairwise edge-set symmetric differences—i.e., diversity. We formally define the DNT problem and introduce the novel concept of Bi-Criteria Triangulation (BCT). Both problems are proven NP-hard. We design a polynomial-time $mathrm{poly}(n,k)$-algorithm achieving a diversity approximation ratio of $1 - Theta(1/k)$. For the minimum pairwise diversity variant, we present an $n^{O(k)}$-time algorithm with a $1/2$-approximation guarantee—substantially outperforming brute-force enumeration. Our approach integrates computational geometry, combinatorial optimization, and structural analysis of Catalan-related triangulation spaces.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We initiate the study of computing diverse triangulations to a given polygon. Given a simple $n$-gon $P$, an integer $ k geq 2 $, a quality measure $sigma$ on the set of triangulations of $P$ and a factor $ alpha geq 1 $, we formulate the Diverse and Nice Triangulations (DNT) problem that asks to compute $k$ emph{distinct} triangulations $T_1,dots,T_k$ of $P$ such that a) their diversity, $sum_{i<j} d(T_i,T_j) $, is as large as possible emph{and} b) they are nice, i.e., $sigma(T_i) leq alpha sigma^* $ for all $1leq i leq k$. Here, $d$ denotes the symmetric difference of edge sets of two triangulations, and $sigma^*$ denotes the best quality of triangulations of $P$, e.g., the minimum Euclidean length. As our main result, we provide a $mathrm{poly}(n,k)$-time approximation algorithm for the DNT problem that returns a collection of $k$ distinct triangulations whose diversity is at least $1 - Theta(1/k)$ of the optimal, and each triangulation satisfies the quality constraint. This is accomplished by studying emph{bi-criteria triangulations} (BCT), which are triangulations that simultaneously optimize two criteria, a topic of independent interest. We complement our approximation algorithms by showing that the DNT problem and the BCT problem are NP-hard. Finally, for the version where diversity is defined as $min_{i<j} d(T_i,T_j) $, we show a reduction from the problem of computing optimal Hamming codes, and provide an $n^{O(k)}$-time $ frac12$-approximation algorithm. Note that this improves over the naive brutef-orce $2^{O(nk)}$ time bound for enumerating all $k$-tuples among the triangulations of a simple $n$-gon, whose total number can be the $(n-2)$-th Catalan number.

Problem

• Computing diverse triangulations for given polygons.
• Optimizing triangulation quality and diversity simultaneously.
• Solving NP-hard problems in poly(n,k)-time approximation.

Innovation

• Poly-time approximation for diverse triangulations
• Bi-criteria optimization for triangulation quality
• NP-hard proof for diverse triangulation problem