Trimmed Moebius Inversion and Graphs of Bounded Degree

TL;DR

This paper introduces a trimmed Moebius inversion technique to efficiently compute combinatorial properties and graph parameters like domatic and chromatic numbers on bounded degree graphs, improving computational bounds.

Contribution

The paper develops a novel trimmed Moebius inversion method that accelerates calculations of set family properties and applies it to optimize algorithms for graph problems with bounded degree.

Findings

Efficient computation of packings, coverings, and partitions within polynomial factors.

Improved algorithms for Domatic Number and Chromatic Number with exponential speedup.

Polynomial factor bounds for graphs with fixed maximum degree.

Abstract

We study ways to expedite Yates's algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an $n$-element universe $U$ and a family $\scr F$ of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of $U$ with $k$ sets from $\scr F$ in time within a polynomial factor (in $n$) of the number of supersets of the members of $\scr F$. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs of maximum degree $\Delta$. In particular, we show how to compute the Domatic Number in time within a polynomial factor of $(2^{\Delta+1-2)^{n/(\Delta+1)$ and the Chromatic Number in time within a polynomial factor of $(2^{\Delta+1-\Delta-1)^{n/(\Delta+1)$. For any constant $\Delta$, these bounds are $O\bigl((2-\epsilon)^n\bigr)$ for $\epsilon>0$ independent of the number of vertices $n$.