Glauber dynamics finds large independent sets in complex graphs efficiently
Large independent sets via neighborhood block dynamics, Glauber dynamics, and local occupancy
Data Structures and Algorithms
Summary
Finding large independent sets in graphs is a common problem with applications in networking and scheduling. The authors improve previous methods by designing short sequences of updates that quickly find such sets even in complicated graphs with certain structural limits. Their work shows that these sequences can handle large graphs efficiently and find sets close to the best known theoretical sizes. They also improve bounds on graph coloring numbers, which relate to how complicated the graph connections are.
What this means in practice
- •For network schedulers: Optimize resource allocation by efficiently finding large independent sets in complex network graphs with structural constraints.
- •For graph algorithm developers: Implement faster algorithms for sampling and coloring large sparse graphs using improved update sequences based on Glauber dynamics.
Authors
Ewan Davies, Brian Tan
Abstract
Local occupancy can be used to bound the expected size of a hard-core independent set. The framework of locally stationary distributions shows that for triangle-free graphs, despite slow mixing at the relevant fugacity parameter, standard Glauber dynamics can be used to find sets close to the local occupancy guarantee. We improve the application of this framework to the hard-core model by replacing a conditional mixing argument with a more direct approach based on a short sequence of single-site updates that, from every initial configuration, achieves the required occupancy inequality. In general graphs, naive conditional mixing estimates or certain problematic structures in the neighborhood can in principle incur running time cost exponential in the maximum degree $Δ$. We design short update sequences that, via our relaxed criterion, yield running times polynomial in $Δ$ for graphs with a hereditary bound $b$ on the density of $K_{t,t}$ subgraphs in neighborhoods. Our methods handle the case where $t$ and $b$ can grow with the maximum degree, and the resulting algorithm attains the random graph shattering threshold when they grow sufficiently slowly. That is, we give improved local occupancy parameters and a short update sequence analysis which together show that Glauber dynamics finds independent sets of size $(1-o(1))n(\logΔ)/Δ$ in time polynomial in $n$ and $Δ$ whenever $t$ and $b$ are constant or grow suitably slowly. In a similar regime, we also establish that the (correspondence) chromatic number is at most $(1+o(1))Δ/\logΔ$ which improves upon prior results by a factor of at least $4$ in the case $t\ge 4$.