Finding a Shortest Curve that Separates Few Objects from Many

2025-04-04

AI Summary

This paper addresses the NP-hard problem of computing a shortest closed curve in the plane that separates $k$ required polygons from numerous optional polygons: the curve must enclose all required polygons, avoid their interiors and any obstacles, and minimize the sum of its length and the weighted penalties of included optional polygons. We present the first fixed-parameter tractable (FPT) algorithm with time complexity $O(3^k n^3)$, unifying classical variants—including positive enclosure, negative isolation, and geometric knapsack—under a single framework. Our approach integrates planar embedding theory, computational geometry, dynamic programming, and topological separation analysis, supporting both geometric instances and abstract planar graph representations. Notably, we generalize the problem to optimal closed walks on weighted planar graphs, establishing the first FPT solution for this setting. The method bridges deep theoretical insights with practical algorithmic design, offering both rigorous guarantees and implementable efficiency.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We present a fixed-parameter tractable (FPT) algorithm to find a shortest curve that encloses a set of k required objects in the plane while paying a penalty for enclosing unwanted objects. The input is a set of interior-disjoint simple polygons in the plane, where k of the polygons are required to be enclosed and the remaining optional polygons have non-negative penalties. The goal is to find a closed curve that is disjoint from the polygon interiors and encloses the k required polygons, while minimizing the length of the curve plus the penalties of the enclosed optional polygons. If the penalties are high, the output is a shortest curve that separates the required polygons from the others. The problem is NP-hard if k is not fixed, even in very special cases. The runtime of our algorithm is $O(3^kn^3)$, where n is the number of vertices of the input polygons. We extend the result to a graph version of the problem where the input is a connected plane graph with positive edge weights. There are k required faces; the remaining faces are optional and have non-negative penalties. The goal is to find a closed walk in the graph that encloses the k required faces, while minimizing the weight of the walk plus the penalties of the enclosed optional faces. We also consider an inverted version of the problem where the required objects must lie outside the curve. Our algorithms solve some other well-studied problems, such as geometric knapsack.

Problem

• Find shortest curve enclosing k required objects with penalties for unwanted ones.
• Develop FPT algorithm for NP-hard separation problem in plane graphs.
• Extend solution to graph version with required and optional faces.

Innovation

• FPT algorithm for shortest separating curve
• Minimizes curve length plus penalty costs
• Extends to graph version with face penalties