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Sampling List Packings

Evan Camrud, Ewan Davies, Alex Karduna, Holden Lee

TL;DR

This work studies the problem of approximately counting $L$-packings, a list-packing analogue of list coloring, on graphs of maximum degree $Δ$ with list size $q$. The authors design and analyze the heat-bath Glauber dynamics for list packings and prove rapid mixing via path coupling, leveraging a novel coupling of perfect matchings in auxiliary availability graphs to handle the many-spin-per-vertex structure ($q!$ permutations per vertex). They establish an FPRAS for the number of $L$-packings when $q \ge CΔ^2$ for a universal constant $C$, under fixed $Δ$ and $q$ in the analysis. This advances algorithmic counting for a structured, high-spin spin system and highlights the role of coupling in overcoming the combinatorial complexity of ensuring disjointness among packings. The results open avenues for tighter bounds on $q$ and for exploring alternative Markov chains and perfect-sampling techniques in the list-packing context.

Abstract

We study the problem of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted a lot of attention. For list packing the setup is similar but we seek a full decomposition of the lists of colors into pairwise-disjoint proper list colorings. In particular, the existence of a list packing implies the existence of a list coloring. Recent works on list packing have focused on existence or extremal results of on the number of list packings, but here we turn to the algorithmic aspects of counting. In graphs of maximum degree $Δ$ and when the number of colors is at least $Ω(Δ^2)$, we give an FPRAS based on rapid mixing of a natural Markov chain (the Glauber dynamics) which we analyze with the path coupling technique. Some motivation for our work is the investigation of an atypical spin system, one where the number of spins for each vertex is much larger than the graph degree.

Sampling List Packings

TL;DR

This work studies the problem of approximately counting -packings, a list-packing analogue of list coloring, on graphs of maximum degree with list size . The authors design and analyze the heat-bath Glauber dynamics for list packings and prove rapid mixing via path coupling, leveraging a novel coupling of perfect matchings in auxiliary availability graphs to handle the many-spin-per-vertex structure ( permutations per vertex). They establish an FPRAS for the number of -packings when for a universal constant , under fixed and in the analysis. This advances algorithmic counting for a structured, high-spin spin system and highlights the role of coupling in overcoming the combinatorial complexity of ensuring disjointness among packings. The results open avenues for tighter bounds on and for exploring alternative Markov chains and perfect-sampling techniques in the list-packing context.

Abstract

We study the problem of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted a lot of attention. For list packing the setup is similar but we seek a full decomposition of the lists of colors into pairwise-disjoint proper list colorings. In particular, the existence of a list packing implies the existence of a list coloring. Recent works on list packing have focused on existence or extremal results of on the number of list packings, but here we turn to the algorithmic aspects of counting. In graphs of maximum degree and when the number of colors is at least , we give an FPRAS based on rapid mixing of a natural Markov chain (the Glauber dynamics) which we analyze with the path coupling technique. Some motivation for our work is the investigation of an atypical spin system, one where the number of spins for each vertex is much larger than the graph degree.
Paper Structure (9 sections, 12 theorems, 24 equations)

This paper contains 9 sections, 12 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Technical overview
  4. Coupling perfect matchings
  5. Organization
  6. Glauber dynamics for list packing
  7. Coupling matchings
  8. Concluding remarks
  9. Acknowledgement

Key Result

Theorem 2.1

There is an absolute constant $C$ such that the following holds. For any $\Delta\ge 1$ and $q\ge C\Delta^2$, let $G$ be a graph of maximum degree $\Delta$. Then for any $q$-list assignment $L$ of $G$, there is an FPRAS for the number of $L$-packings of $G$.

Theorems & Definitions (20)

  • Theorem 2.1
  • Lemma 2.1
  • Theorem 3.1: Bubley and Dyer BD97, see also DG99
  • Lemma 3.2: See e.g., Jer03
  • Lemma 3.2
  • Lemma 4.1: Corollary of Hall's theorem PHal35
  • proof : Sketch proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 10 more