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On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints

Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, Abhishek Sahu

TL;DR

It is proved that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A| and |A| and the vertex cover number (vc) is considered as the parameter and a kernel with O(vc^2) vertices is provided.

Abstract

Given an undirected graph G and a set A \subseteq V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G) \setminus A. An A-path is called an (A,\ell)-path if the length of the path is exactly \ell. In the {\sc (A, \ell)-Path Packing} problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, \ell)-paths in G or not. We pursue this problem with respect to structural parameters. We prove that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A|. In addition, we consider the combined parameters distance to cluster (cvd) + |A| and distance to cluster (cvd) + \ell. For both these combined parameters, we provide FPT algorithms. Finally, we consider the vertex cover number (vc) as the parameter and provide a kernel with O(vc^2) vertices.

On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints

TL;DR

It is proved that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A| and |A| and the vertex cover number (vc) is considered as the parameter and a kernel with O(vc^2) vertices is provided.

Abstract

Given an undirected graph G and a set A \subseteq V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G) \setminus A. An A-path is called an (A,\ell)-path if the length of the path is exactly \ell. In the {\sc (A, \ell)-Path Packing} problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, \ell)-paths in G or not. We pursue this problem with respect to structural parameters. We prove that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A|. In addition, we consider the combined parameters distance to cluster (cvd) + |A| and distance to cluster (cvd) + \ell. For both these combined parameters, we provide FPT algorithms. Finally, we consider the vertex cover number (vc) as the parameter and provide a kernel with O(vc^2) vertices.
Paper Structure (10 sections, 13 theorems, 4 figures)

This paper contains 10 sections, 13 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Algorithmic lower bound based on rETH when parameterized by ${\normalfont \textsf{dtp}} + |A|$
  4. Essential results
  5. Hardness proof
  6. Construction Phase $2$.
  7. FPT algorithm parameterized by ${\normalfont \textsf{cvd}} + |A|$
  8. Phase 1: The Guessing Phase
  9. Phase 2: Bounding the number of Cliques Phase
  10. FPT algorithm when parameterized by ${\normalfont \textsf{cvd}} + \ell$

Key Result

Theorem 1

Unless Conjecture conj:rETH fails, there is no randomized algorithm for $(A, \ell)$-Path Packing that runs in $f({\normalfont \textsf{dtp}} +|A|)\cdot n^{o(\sqrt{{\normalfont \textsf{dtp}} +|A|})}$-time and correctly decides with probability at least $2/3$ when ${\normalfont \textsf{dtp}}$ is the di

Figures (4)

  • Figure 1: Structural Parameterizations of ALPP. The arrow represents the hierarchy of different structural parameters, while the dashed line represents the parameters that have yet to be explored in the context of our problems.
  • Figure 2: Illustration of Construction Phase 1. Note that $|C_i| = \overline{n_i} = 8nN - n_i$.
  • Figure 3: Illustration of construction of $(A, \ell)$-Path Packing. Note that $|C_i| = \overline{n_i} = 8nN - n_i$.
  • Figure 4: Construction of the paths $\lambda_i$ in the proof of Lemma \ref{['lemma:easypart']}.

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 1: Separation Lemma
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5: New $q$-Expansion Lemma, Babu0R22FominLLSTZ19JacobMR23
  • Definition 6: Fixed-Parameter Tractability
  • Definition 7: Kernelization
  • Conjecture 8: rETH
  • Proposition 8
  • ...and 6 more