A Dynamic Programming Framework for Generating Approximately Diverse and Optimal Solutions

2025-01-21

AI Summary

This work addresses NP-hard geometric and combinatorial optimization problems—including polygon triangulation, planar graph independent set, and vertex cover—by proposing the Approximate Diversified Dynamic Programming (ADDP) framework. The goal is to efficiently generate $k$ solutions that are simultaneously $c$-approximate optimal and maximally diverse with respect to symmetric difference distance. For polygon triangulation, we present the first polynomial-time algorithm achieving $mathrm{poly}(n,k)$ time complexity and a $(1 - Theta(1/k))$-approximation guarantee. For diversified planar independent set and vertex cover, we give the first PTAS: in $2^{O(kdelta^{-1}varepsilon^{-2})} n^{O(1/varepsilon)}$ time, it outputs solutions that are $(1-varepsilon)$-diverse and $(1-delta)c$-optimal. Our approach integrates dynamic programming, planar graph divide-and-conquer, parameterized diversity optimization, and enumeration over $c$-approximate solution spaces—systematically advancing both the theoretical foundations and practical applicability of diversified optimization for geometric problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We develop a general framework, called approximately-diverse dynamic programming (ADDP) that can be used to generate a collection of $kge2$ maximally diverse solutions to various geometric and combinatorial optimization problems. Given an approximation factor $0le cle1$, this framework also allows for maximizing diversity in the larger space of $c$-approximate solutions. We focus on two geometric problems to showcase this technique: 1. Given a polygon $P$, an integer $kge2$ and a value $cle1$, generate $k$ maximally diverse $c$-nice triangulations of $P$. Here, a $c$-nice triangulation is one that is $c$-approximately optimal with respect to a given quality measure $sigma$. 2. Given a planar graph $G$, an integer $kge2$ and a value $cle1$, generate $k$ maximally diverse $c$-optimal Independent Sets (or, Vertex Covers). Here, an independent set $S$ is said to be $c$-optimal if $|S|ge c|S'|$ for any independent set $S'$ of $G$. Given a set of $k$ solutions to the above problems, the diversity measure we focus on is the average distance between the solutions, where $d(X,Y)=|XDelta Y|$. For arbitrary polygons and a wide range of quality measures, we give $ ext{poly}(n,k)$ time $(1-Theta(1/k))$-approximation algorithms for the diverse triangulation problem. For the diverse independent set and vertex cover problems on planar graphs, we give an algorithm that runs in time $2^{O(kdelta^{-1}epsilon^{-2})}n^{O(1/epsilon)}$ and returns $(1-epsilon)$-approximately diverse $(1-delta)c$-optimal independent sets or vertex covers. Our triangulation results are the first algorithmic results on computing collections of diverse geometric objects, and our planar graph results are the first PTAS for the diverse versions of any NP-complete problem. Additionally, we also provide applications of this technique to diverse variants of other geometric problems.

Problem

• Diversity Optimization
• Polygon Triangulation
• Graph Independent Set

Innovation

• Approximate Diversity Dynamic Programming
• Polygon Triangulation
• Combinatorial Optimization