Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Simpler Approach for Monotone Parametric Minimum Cut: Finding the Breakpoints in Order

Arne Beines, Michael Kaibel, Philip Mayer, Petra Mutzel, Jonas Sauer

Abstract

We present parametric breadth-first search (PBFS), a new algorithm for solving the parametric minimum cut problem in a network with source-sink-monotone capacities. The objective is to find the set of breakpoints, i.e., the points at which the minimum cut changes. It is well known that this problem can be solved in the same asymptotic runtime as the static minimum cut problem. However, existing algorithms that achieve this runtime bound involve fairly complicated steps that are inefficient in practice. PBFS uses a simpler approach that discovers the breakpoints in ascending order, which allows it to achieve the desired runtime bound while still performing well in practice. We evaluate our algorithm on benchmark instances from polygon aggregation and computer vision. Polygon aggregation was recently proposed as an application for parametric minimum cut, but the monotonicity property has not been exploited fully. PBFS outperforms the state of the art on most benchmark instances, usually by a factor of 2-3. It is particularly strong on instances with many breakpoints, which is the case for polygon aggregation. Compared to the existing min-cut-based approach for polygon aggregation, PBFS scales much better with the instance size. On large instances with millions of vertices, it is able to compute all breakpoints in a matter of seconds.

A Simpler Approach for Monotone Parametric Minimum Cut: Finding the Breakpoints in Order

Abstract

We present parametric breadth-first search (PBFS), a new algorithm for solving the parametric minimum cut problem in a network with source-sink-monotone capacities. The objective is to find the set of breakpoints, i.e., the points at which the minimum cut changes. It is well known that this problem can be solved in the same asymptotic runtime as the static minimum cut problem. However, existing algorithms that achieve this runtime bound involve fairly complicated steps that are inefficient in practice. PBFS uses a simpler approach that discovers the breakpoints in ascending order, which allows it to achieve the desired runtime bound while still performing well in practice. We evaluate our algorithm on benchmark instances from polygon aggregation and computer vision. Polygon aggregation was recently proposed as an application for parametric minimum cut, but the monotonicity property has not been exploited fully. PBFS outperforms the state of the art on most benchmark instances, usually by a factor of 2-3. It is particularly strong on instances with many breakpoints, which is the case for polygon aggregation. Compared to the existing min-cut-based approach for polygon aggregation, PBFS scales much better with the instance size. On large instances with millions of vertices, it is able to compute all breakpoints in a matter of seconds.

Paper Structure

This paper contains 20 sections, 26 theorems, 4 equations, 2 figures, 6 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constant Capacities
  4. Parametric Capacities
  5. Related Work
  6. Static Max-Flow
  7. Monotone Parametric Max-Flow
  8. Applications
  9. Parametric BFS
  10. Proof of Correctness and Runtime
  11. Overview
  12. Source Contraction
  13. Residual Edges and Sink Component Tree
  14. Flow and Excess Functions
  15. Extending the Flow
  16. ...and 5 more sections

Key Result

Corollary 2

For any flow network $G$, there is a unique sink-minimal minimum cut $C=(V_s,V_t)$, which equals $C(f)$ for every maximum flow $f$.

Figures (2)

  • Figure 1: Left: A polygon aggregation instance with input polygons in gray, a triangulation in white and a possible solution delineated in blue. Right: The corresponding flow network, in which vertices represent faces and edges represent boundaries. Source component vertices in black; sink component vertices in blue; cut edges in orange.
  • Figure 2: Performance of DS-IBFS on aggregation instances, depending on the approximation factor $\varepsilon=10^{-x}$. Left: Ratio of found breakpoints, compared to $\varepsilon=0$. Right: Speedup of PBFS over DS-IBFS.

Theorems & Definitions (26)

  • Corollary 2
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Corollary 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • Lemma 12
  • ...and 16 more