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Finding a Shortest Curve that Separates Few Objects from Many

Therese Biedl, Éric Colin de Verdière, Fabrizio Frati, Anna Lubiw, Günter Rote

TL;DR

The paper introduces Enclosure-with-Penalties, an FPT framework for finding a shortest curve that encloses a subset of objects while paying penalties for enclosing others, in both geometric and graph settings. It develops a dynamic-programming approach on free-space edges, plus an uncrossing step to produce a valid weakly simple cycle, with a Dijkstra-style optimization that achieves a runtime of $O(3^k n^3)$ and space $O(2^k n^2)$. The method hinges on winding numbers to certify containment and on transforming graph instances to a geometric embedding, enabling a unified treatment and enabling reductions to geometric knapsack and related problems. Exponential-time lower bounds under ETH show the parameter dependence on $k$ is essentially tight. The work also discusses inverted formulations, negative penalties, and broad extensions, offering a versatile toolkit for geometric and graph enclosure problems with penalties.

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.

Finding a Shortest Curve that Separates Few Objects from Many

TL;DR

The paper introduces Enclosure-with-Penalties, an FPT framework for finding a shortest curve that encloses a subset of objects while paying penalties for enclosing others, in both geometric and graph settings. It develops a dynamic-programming approach on free-space edges, plus an uncrossing step to produce a valid weakly simple cycle, with a Dijkstra-style optimization that achieves a runtime of and space . The method hinges on winding numbers to certify containment and on transforming graph instances to a geometric embedding, enabling a unified treatment and enabling reductions to geometric knapsack and related problems. Exponential-time lower bounds under ETH show the parameter dependence on is essentially tight. The work also discusses inverted formulations, negative penalties, and broad extensions, offering a versatile toolkit for geometric and graph enclosure problems with penalties.

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 , 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.

Paper Structure

This paper contains 56 sections, 24 theorems, 18 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Geometric-Enclosure-with-Penalties.
  3. Graph-Enclosure-with-Penalties.
  4. A common framework.
  5. Our approach.
  6. Lower bounds.
  7. Swapping the inside with the outside.
  8. Negative penalties.
  9. Related work
  10. Geometric knapsack problem.
  11. Relation to homotopy and homology.
  12. Specifying only the number of objects to be enclosed.
  13. More variations.
  14. Preliminaries
  15. Weakly simple polygons
  16. ...and 41 more sections

Key Result

Theorem 1

Geometric-Enclosure-with-Penalties for $k$ required polygons can be solved in $O(3^kn^3)$ time and $O(2^kn^2)$ space, if the input polygons have $n$ vertices in total.

Figures (14)

  • Figure 1: The Geometric-Enclosure-with-Penalties problem. The weakly simple polygon $W$ (in red) encloses the required polygons $R$ (yellow hatched) while paying a penalty for enclosing any of the optional polygons $O$ (in gray, darker for larger penalties). Overlapping paths are slightly displaced for visibility. Changing $W$ via the dashed path to put polygon $P$ outside would be preferable if the increase in length is less than the penalty of $P$.
  • Figure 2: The Graph-Enclosure-with-Penalties problem. The colors have the same meaning as in \ref{['fig:example-geom']}. A weakly simple closed walk $W$ which is a solution for the instance is red (bold). The edge weights, which are not shown, are unrelated to the Euclidean lengths.
  • Figure 3: (a) A weakly simple polygon drawn via its $\varepsilon$-approximation. (b) The edges, after subdividing at interior vertices (forks), are partitioned into interior faces. Four faces are corridors and five are chambers. The largest chamber (in yellow) is almost-simple but not simple. (c) Non-uniqueness of the faces for a weakly simple polygon that traverses a line segment four times. In the top figure the two vertices on the left are transition vertices; this is reversed in the bottom figure.
  • Figure 4: Cases of the recursion. Solid edges are free-space edges; dashed edges are mouths.
  • Figure 5: Gluing together closed walks, which may cross each other and are possibly self-crossing.
  • ...and 9 more figures

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Lemma 5
  • Proposition 5: Uncrossing Eulerian plane multigraphs
  • Lemma 6
  • Definition 7
  • Lemma 8
  • ...and 19 more