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Faster algorithms for graph homomorphism via tractable constraint satisfaction

Clément Carbonnel

Abstract

We show that the existence of a homomorphism from an $n$-vertex graph $G$ to an $h$-vertex graph $H$ can be decided in time $2^{O(n)}h^{O(1)}$ and polynomial space if $H$ comes from a family of graphs that excludes a topological minor. The algorithm is based on a reduction to a single-exponential number of constraint satisfaction problems over tractable languages and can handle cost minimization. We also present an improved randomized algorithm for the special case where the graph $H$ is an odd cycle.

Faster algorithms for graph homomorphism via tractable constraint satisfaction

Abstract

We show that the existence of a homomorphism from an -vertex graph to an -vertex graph can be decided in time and polynomial space if comes from a family of graphs that excludes a topological minor. The algorithm is based on a reduction to a single-exponential number of constraint satisfaction problems over tractable languages and can handle cost minimization. We also present an improved randomized algorithm for the special case where the graph is an odd cycle.
Paper Structure (13 sections, 20 theorems, 14 equations, 1 figure)

This paper contains 13 sections, 20 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Preliminaries
  4. Graphs
  5. Minors and topological minors.
  6. Tree decompositions.
  7. Layerings.
  8. Track layouts.
  9. Homomorphism problems.
  10. Valued constraint satisfaction problems
  11. Persistent majority colorings
  12. Excluding a topological minor
  13. Odd cycle homomorphism

Key Result

Theorem 1

If $\cal H$ is a family of graphs that excludes at least one topological minor, then $\textsc{ValHom}(\cal H)$ can be solved in time $2^{O(n)} (h+|\eta|)^{O(1)}$ and polynomial space.

Figures (1)

  • Figure 1: The sets $S_4$, $S_4^A$ and $S_4^B$ (from left to right) for $C_9$.

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Theorem 3: Thapper and Živný DBLP:journals/siamcomp/ThapperZ17
  • Definition 3
  • Theorem 4
  • proof
  • Lemma 1
  • proof
  • ...and 43 more