Improved Deterministic Distributed Maximum Weight Independent Set Approximation in Sparse Graphs

Abstract

We design new deterministic CONGEST approximation algorithms for maximum weight independent set (MWIS) in 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(log² (Δ W)⋅ log (1/ε)+log ^{*}n)-round algorithm, where W is the largest node-weight (this bound translates to O(log² n⋅log (1/ε)) under the common assumption that W = poly(n)). As for α-dependent approximations, a deterministic CONGEST (8(1+ε)⋅α)-approximation algorithm with runtime O(log³ n⋅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]. As our main results, we show the following.  
- A deterministic CONGEST algorithm that computes an α^{1+τ}-approximation for MWIS in O(log nlog α) rounds for any constant τ > 0. To the best of our knowledge, this is the fastest runtime of any deterministic non-trivial approximation algorithm for MWIS to date. Furthermore, for the large class of graphs where α = Δ^{1-Θ(1)}, it implies a deterministic Δ^{1-Θ(1)}-approximation algorithm with a runtime of O(log nlog α) which improves upon the result of Faour et al. in both approximation ratio (by a Δ^{Θ(1)} factor) and runtime (by an O(log n/log α) factor). 
- A deterministic CONGEST algorithm that computes an O(α)-approximation for MWIS in O(α^{τ}log n) rounds for any (desirably small) constant τ > 0. This improves the runtime of the best known deterministic O(α)-approximation algorithm in the case that α = O(polylog n). This also leads to a deterministic Δ(1+ε)-approximation algorithm with a runtime of O(α^{τ}log nlog (1/ε)) which improves upon the runtime of Faour et al. in the case that α = O(polylog n). 
- A deterministic CONGEST algorithm that computes a (⌊(2+ε)α⌋)-approximation for MWIS in O(αlog n) rounds. This improves upon the best known α-dependent approximation ratio by a constant factor. 
- A deterministic CONGEST algorithm that computes a 2d²-approximation for MWIS in time O(d²+log ^{*}n) in a directed graph with out-degree at most d. The dependency on n is (asymptotically) optimal due to a lower bound by Czygrinow et al. [DISC 2008] and Lenzen and Wattenhofer [DISC 2008].  We note that a key ingredient to all of our algorithms is a novel deterministic method that computes a high-weight subset of nodes whose induced subgraph is sparse.