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Fixed-Parameter Tractability of Hedge Cut

Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Daniel Lokshtanov, Saket Saurabh

TL;DR

This paper shows that Hedge Cut is fixed-parameter tractable parameterized by the solution size $\ell$ by providing an algorithm with running time $\binom{O(\log n) + \ell}{\ell} \cdot m^{O(1)}$, which can be upper bounded by $c^{\ell} \cdot (n+m)^{O(1)}$ for any constant $c>1$.

Abstract

In the Hedge Cut problem, the edges of a graph are partitioned into groups called hedges, and the question is what is the minimum number of hedges to delete to disconnect the graph. Ghaffari, Karger, and Panigrahi [SODA 2017] showed that Hedge Cut can be solved in quasipolynomial-time, raising the hope for a polynomial time algorithm. Jaffke, Lima, Masarík, Pilipczuk, and Souza [SODA 2023] complemented this result by showing that assuming the Exponential Time Hypothesis (ETH), no polynomial-time algorithm exists. In this paper, we show that Hedge Cut is fixed-parameter tractable parameterized by the solution size $\ell$ by providing an algorithm with running time $\binom{O(\log n) + \ell}{\ell} \cdot m^{O(1)}$, which can be upper bounded by $c^{\ell} \cdot (n+m)^{O(1)}$ for any constant $c>1$. This running time captures at the same time the fact that the problem is quasipolynomial-time solvable, and that it is fixed-parameter tractable parameterized by $\ell$. We further generalize this algorithm to an algorithm with running time $\binom{O(k \log n) + \ell}{\ell} \cdot n^{O(k)} \cdot m^{O(1)}$ for Hedge $k$-Cut.

Fixed-Parameter Tractability of Hedge Cut

TL;DR

This paper shows that Hedge Cut is fixed-parameter tractable parameterized by the solution size by providing an algorithm with running time , which can be upper bounded by for any constant .

Abstract

In the Hedge Cut problem, the edges of a graph are partitioned into groups called hedges, and the question is what is the minimum number of hedges to delete to disconnect the graph. Ghaffari, Karger, and Panigrahi [SODA 2017] showed that Hedge Cut can be solved in quasipolynomial-time, raising the hope for a polynomial time algorithm. Jaffke, Lima, Masarík, Pilipczuk, and Souza [SODA 2023] complemented this result by showing that assuming the Exponential Time Hypothesis (ETH), no polynomial-time algorithm exists. In this paper, we show that Hedge Cut is fixed-parameter tractable parameterized by the solution size by providing an algorithm with running time , which can be upper bounded by for any constant . This running time captures at the same time the fact that the problem is quasipolynomial-time solvable, and that it is fixed-parameter tractable parameterized by . We further generalize this algorithm to an algorithm with running time for Hedge -Cut.

Paper Structure

This paper contains 7 sections, 8 theorems, 5 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithm
  4. Description of the algorithm
  5. Analysis of the algorithm
  6. Hardness
  7. Conclusion

Key Result

Theorem 1

There is a $\binom{\mathcal{O}(k \log n) + \ell}{\ell} \cdot n^{\mathcal{O}(k)} \cdot m^{\mathcal{O}(1)}$ time randomized algorithm with one-sided error for Hedge $k$-Cut, where $\ell$ is the size of an optimum hedge $k$-cut-set, $n$ is the number of vertices, and $m$ is the number of hedges.

Figures (1)

  • Figure 1: Example of a hedge graph on 6 vertices. It has two hedges, yellow and red. Both hedges are of size 6. The red hedge consists of two components, each containing 3 elements. The yellow hedge has 3 components, each of size 2. The hedge 2-cut-set associated with a 2-cut $(V_1, V_2)$ consists of all components of the yellow hedge.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 2 more