On Reconstructing a Convex Polygon from Partial Information

2026-07-01

AI Summary

This study investigates the feasibility and algorithmic approaches for reconstructing convex polygons from partial geometric information, such as ordered sets of edge lengths or interior angles. By systematically analyzing various combinations of input features, the work provides the first comprehensive characterization of reconstructability under diverse conditions. Integrating techniques from computational geometry, algorithm design, and complexity theory, the authors develop several efficient algorithms for both decision and reconstruction tasks, while also establishing NP-hardness for certain cases. The paper establishes a unified theoretical framework that not only clarifies the boundaries of tractability but also identifies multiple open problems, thereby laying a foundation for future research in polygon reconstruction.

Abstract

The reconstruction problem asks to construct a (convex) polygon that has a specified set of features, such as an ordered set of edge-lengths or an ordered set of polygon-angles. In this paper, we do a systematic exploration of the reconstruction problem in all scenarios where one or two sets of features have been specified. Some of these scenarios were well-studied already, for some we develop testing-algorithms and/or hardness results, and many give rise to interesting open problems for future study.

Problem

• convex polygon
• reconstruction problem
• edge-lengths
• polygon-angles
• partial information

Innovation

• convex polygon reconstruction
• partial information
• feature specification
• algorithmic hardness
• computational geometry