Improved Deterministic Distributed Maximum Weight Independent Set Approximation in Sparse Graphs
Yuval Gil
TL;DR
This work delivers deterministic CONGEST algorithms for MWIS in sparse graphs by introducing Sparse_Set, a primal-dual–driven procedure powered by β-bounded colorings and arbdefective colorings. The authors obtain a spectrum of approximation guarantees tied to arboricity α and, separately, to maximum degree Δ, including Δ(1+ε)-, ⌊(2+ε)·α⌋-, and α^{1+τ}-type approximations, with runtimes ranging from O(α log n) to O(log α log n) in various regimes. They also extend the approach to directed graphs with out-degree d, achieving a 2d²-approximation in O(d²+log^* n) rounds. Overall, the results advance deterministic MWIS approximations in CONGEST for bounded-arboricity families, offering practical benefits for planar, minor-free, and similarly sparse networks.
Abstract
We design new deterministic CONGEST approximation algorithms for \emph{maximum weight independent set (MWIS)} in \emph{sparse graphs}. As our main results, we obtain new $Δ(1+ε)$-approximation algorithms as well as algorithms whose approximation ratio depend strictly on $α$, in graphs with maximum degree $Δ$ and arboricity $α$. For (deterministic) $Δ(1+ε)$-approximation, the current state-of-the-art is due to a recent breakthrough by Faour et al.\ [SODA 2023] that showed an $O(\log^{2} (ΔW)\cdot \log (1/ε)+\log ^{*}n)$-round algorithm, where $W$ is the largest node-weight (this bound translates to $O(\log^{2} n\cdot\log (1/ε))$ under the common assumption that $W=\text{poly}(n)$). As for $α$-dependent approximations, a deterministic CONGEST $(8(1+ε)\cdotα)$-approximation algorithm with runtime $O(\log^{3} n\cdot\log (1/ε))$ can be derived by combining the aforementioned algorithm of Faour et al.\ with a method presented by Kawarabayashi et al.\ [DISC 2020].