Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Packing Short Cycles

Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, William Lochet, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, Kirill Simonov

TL;DR

The Min-Sum Cycle Packing problem is proved to be Fixed-Parameter Tractable (FPT) when parameterized by k on weighted planar graphs and a polynomial kernel is obtained for the edge-disjoint variant of the problem on planar graphs.

Abstract

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, $k$ vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed $k$, solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by $k$. The same results hold for the version of the problem where the task is to find $k$ edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of $k$ shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by $k$ on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions.

Packing Short Cycles

TL;DR

The Min-Sum Cycle Packing problem is proved to be Fixed-Parameter Tractable (FPT) when parameterized by k on weighted planar graphs and a polynomial kernel is obtained for the edge-disjoint variant of the problem on planar graphs.

Abstract

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed , solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by . The same results hold for the version of the problem where the task is to find edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions.

Paper Structure

This paper contains 18 sections, 30 theorems, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Remarks on impact of the weight function.
  3. Overview of the methods
  4. Min-Sum Cycle Packing
  5. Cycle packing on planar graphs
  6. Related Work
  7. Preliminaries
  8. Graphs.
  9. Parameterized Complexity.
  10. Min-Sum Cycle Packing is in XP
  11. Lower bound for Min-Sum Cycle Packing
  12. Packings shortest cycles in planar graphs
  13. Laminar decomposition for shortest cycles
  14. FPT algorithm for Shortest Cycle Packing
  15. Kernelization for Edge-Disjoint Shortest Cycle Packing
  16. ...and 3 more sections

Key Result

Theorem 1

Min-Sum Cycle Packing and Min-Sum Edge-Disjoint Cycle Packing can be solved in $n^{\mathcal{O}(k^6)}$ time, where $n$ is the number of vertices in the graph.

Figures (4)

  • Figure 1: A solution $\mathcal{C}=\{C_1,\ldots,C_5\}$ and the tree representing $\mathcal{C}$.
  • Figure 2: An example of the vertex-selection gadget with $\nu = \Delta = 3$ for one color $i$. The chords depict paths of length $2(\nu-1)(3\Delta-1) = 32$ and one of the ways to pick three vertex-disjoint cycles using three chords is highlighted. This solution picks vertex $v^i_3$. Paths encoding two edges incident to $v^i_3$ are shown by dashed and dotted lines, respectively.
  • Figure 3: Construction of $\mathcal{P}_s(C)$ and $\mathcal{C}_s(C)$ for a splittable cycle (a) and construction of $\mathcal{C}_u(C)=\{C_1,C_2,C_3,C_4\}$ for an unsplittable cycle (b).
  • Figure 4: Construction of $\mathcal{T}^*$. In the tree on the right side of the figure, the parent node of $C_2$ and $C_3$ corresponds to the cycle $C'$ (red dashed lines) on the left.

Theorems & Definitions (65)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 2.2: Chen01a
  • Proposition 2.3: Special case of Erdős-Szekeres theorem erdos1935combinatorial
  • Proposition 2.4: FrankT87
  • Lemma 3.2
  • proof
  • Claim 3.3
  • ...and 55 more