Distributed Lovász Local Lemma under Bandwidth Limitations

TL;DR

The work advances distributed LLL under bandwidth constraints by formalizing simulatable CONGEST LLLs and introducing two solver families: disjoint variable set LLLs and binary LLLs with low risk. It develops post-shattering techniques and network-decomposition-based post-processing to solve small components efficiently, enabling polyloglog n-round CONGEST algorithms. These solvers are then applied to subgraph sampling and, notably, to coloring sparse and triangle-free graphs with far fewer colors than in prior LOCAL-model results, achieving exponential speedups. The results rely on careful risk analysis, shattering arguments, and coordinated parallelism to overcome bandwidth limitations, yielding practical, scalable distributed coloring and sampling tools. The findings have broad implications for fast distributed symmetry breaking and subgraph sampling under bandwidth constraints, with potential impact on networked systems and large-scale graph processing.

Abstract

The constructive Lovász Local Lemma has become a central tool for designing efficient distributed algorithms. While it has been extensively studied in the classic LOCAL model that uses unlimited bandwidth, much less is known in the bandwidth-restricted CONGEST model. In this paper, we present bandwidth- and time-efficient algorithms for various subclasses of LLL problems, including a large class of subgraph sampling problems that are naturally formulated as LLLs. Lastly, we use our LLLs to design efficient CONGEST algorithms for coloring sparse and triangle-free graphs with few colors. These coloring algorithms are exponentially faster than previous LOCAL model algorithms.

Figures (2)

• Figure 1: An illustration of the cases a)--d) that can appear in the post-shattering phase of the degree-bounded subgraph problem. Note that the illustration is only schematic and such a tight example with $\Delta=6$ does not satisfy any LLL criterion. The colors represent the variable assignments after the initial sampling. The question mark indicates that the respective variable got retracted and participates in the post-shattering phase. In a), the vertex $v$ got an extremely bad split and retracted all of its incident variables. In b), $v$ is affected by a retraction from a type a) node, and hence retracts all of its incident $\mathsf{white}$ variables. In c), we see a node that was neither unhappy nor affected, but has some of its $\mathsf{white}$ variables retracted by some other node of type c). The node in d) is happy and does not undergo any retractions. It does not participate in the post-shattering phase. W.h.p. the bulk of the nodes are of type d).
• Figure 2: An example of a node $v$ with $\Delta=8$ neighbors and two examples of sampled subsets (red patterned nodes, black nodes). The dashed edges are non-edges, that is, all other edges e.g., the edge $\{1,6\}$ are present in the graph. This neighborhood has $14$ non-edges out of the $\binom{8}{2}=28$ tentative edges. While $v$ has only degree $4=\Delta/2$ into either of the two subsets, the red patterned subset would be a sampled subset with a small sparsity, as it only contains the single non-edge $\{1,8\}$. The black subset has larger sparsity, as it contains five non-edges $\{2,3\}, \{3,4\},\{2,4\}, \{4,6\},$ and $\{3,6\}$. The number of non-edges in a sampled set is not a linear function of the nodes' sampling status.

Theorems & Definitions (72)

• Definition 2.1
• Definition 2.3: simulatable
• Theorem 3.1
• Theorem 3.1
• Definition 4.0
• Definition 4.1: binary LLLs with low risk
• Remark 4.2
• Theorem 4.3
• Lemma 4.4
• proof