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Pattern-Sparse Tree Decompositions in $H$-Minor-Free Graphs

Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk

Abstract

Given an $H$-minor-free graph $G$ and an integer $k$, our main technical contribution is sampling in randomized polynomial time an induced subgraph $G'$ of $G$ and a tree decomposition of $G'$ of width $\widetilde{O}(k)$ such that for every $Z\subseteq V(G)$ of size $k$, with probability at least $\left(2^{\widetilde{O}(\sqrt{k})}|V(G)|^{O(1)}\right)^{-1}$, we have $Z \subseteq V(G')$ and every bag of the tree decomposition contains at most $\widetilde{O}(\sqrt{k})$ vertices of $Z$. Having such a tree decomposition allows us to solve a wide range of problems in (randomized) time $2^{\widetilde{O}(\sqrt{k})}n^{O(1)}$ where the solution is a pattern $Z$ of size $k$, e.g., Directed $k$-Path, $H$-Packing, etc. In particular, our result recovers all the algorithmic applications of the pattern-covering result of Fomin et al. [SIAM J. Computing 2022] (which requires the pattern to be connected) and the planar subgraph-finding algorithms of Nederlof [STOC 2020]. Furthermore, for $K_{h,3}$-free graphs (which include bounded-genus graphs) and for a fixed constant $d$, we signficantly strengthen the result by ensuring that not only $Z$ has intersection $\widetilde{O}(\sqrt{k})$ with each bag, but even the distance-$d$ neighborhood $N^d_{G}[Z]$ as well. This extension makes it possible to handle a wider range of problems where the neighborhood of the pattern also plays a role in the solution, such as partial domination problems and problems involving distance constraints.

Pattern-Sparse Tree Decompositions in $H$-Minor-Free Graphs

Abstract

Given an -minor-free graph and an integer , our main technical contribution is sampling in randomized polynomial time an induced subgraph of and a tree decomposition of of width such that for every of size , with probability at least , we have and every bag of the tree decomposition contains at most vertices of . Having such a tree decomposition allows us to solve a wide range of problems in (randomized) time where the solution is a pattern of size , e.g., Directed -Path, -Packing, etc. In particular, our result recovers all the algorithmic applications of the pattern-covering result of Fomin et al. [SIAM J. Computing 2022] (which requires the pattern to be connected) and the planar subgraph-finding algorithms of Nederlof [STOC 2020]. Furthermore, for -free graphs (which include bounded-genus graphs) and for a fixed constant , we signficantly strengthen the result by ensuring that not only has intersection with each bag, but even the distance- neighborhood as well. This extension makes it possible to handle a wider range of problems where the neighborhood of the pattern also plays a role in the solution, such as partial domination problems and problems involving distance constraints.

Paper Structure

This paper contains 49 sections, 31 theorems, 117 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our results.
  3. Algorithmic applications
  4. Subgraph isomorphism
  5. Steiner trees
  6. Disjoint paths
  7. Densest subgraph
  8. Partial cover problems
  9. Distance constraints
  10. Map graphs
  11. Overview
  12. Duality result
  13. Moving to a sparse separation
  14. The main algorithm
  15. Extending to the bounded-distance version ($d\geqslant 1$)
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a class of graphs that exclude a fixed apex graph as a minor. Then there exists a randomized polynomial-time algorithm that, given an $n$-vertex graph $G$ from $\mathcal{C}$ and an integer $k$, samples a vertex subset $A \subseteq V(G)$ with the following properties:

Figures (6)

  • Figure 1: Separating two sets $J_1$ and $J_h$.
  • Figure 2: If vertex $z$ is adjacent to $h$ internally disjoint $s-t$ paths and it is not on these paths, then there is $K_{3,h}$-minor: we contract the blue subpaths into $s$ and the gray subpaths into $t$. More generally, the same is true if $z$ can reach each $s-t$ path without going through any other $s-t$ paths (paths highlighted in red), in which case we contract these paths into $z$..
  • Figure 3: Contractions to obtain a $K_{3,3}$-minor. The paths highlighted in yellow are contracted into $u_1$, $u_2$, and $u_3$. The pahs highlighted in blue, gray, and red are contracted into $s$, $t$, and $v'$, respectively.
  • Figure 4: Diagram of the control flow of the main procedure.
  • Figure 5: Diagram of the control flow of the non-leaf recursive call.
  • ...and 1 more figures

Theorems & Definitions (109)

  • Theorem 1.1: Fomin et al. DBLP:journals/siamcomp/FominLMPPS22
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Theorem 9 in DBLP:journals/siamcomp/FominLMPPS22
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Theorem 3.1
  • ...and 99 more