[Paper Review] Distributed Approximation of Maximum Independent Set and Maximum Matching

Reuven Bar-Yehuda, Keren Censor-Hillel+2 more|arXiv (Cornell University)|Aug 1, 2017

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper presents efficient distributed algorithms for approximating maximum independent set (MaxIS) and maximum weight matching (MWM) in the CONGEST model. It introduces a deterministic Δ-approximation for MaxIS in O(Δ + log* n) rounds using graph coloring, and a randomized O(log Δ / log log Δ)-round algorithm for (2+ε)-approximate MWM, achieving optimal round complexity with respect to Δ dependency.

ABSTRACT

We present a simple distributed $Δ$-approximation algorithm for maximum weight independent set (MaxIS) in the $\mathsf{CONGEST}$ model which completes in $O( exttt{MIS}(G)\cdot \log W)$ rounds, where $Δ$ is the maximum degree, $ exttt{MIS}(G)$ is the number of rounds needed to compute a maximal independent set (MIS) on $G$, and $W$ is the maximum weight of a node. %Whether our algorithm is randomized or deterministic depends on the exttt{MIS} algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in $O(\log n \log W)$ rounds, where $n$ is the number of nodes. We also present a deterministic $O(Δ+\log^* n)$-round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a $2$-approximation for maximum weight matching without incurring any additional round penalty in the $\mathsf{CONGEST}$ model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of \emph{local aggregation algorithms} for which we describe a mechanism that allows the simulation to run in the $\mathsf{CONGEST}$ model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ($2+\varepsilon$) allows us to devise a distributed algorithm requiring $O(\frac{\log Δ}{\log\logΔ})$ rounds for any constant $\varepsilon>0$. For the unweighted case, we can even obtain a $(1+\varepsilon)$-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on $Δ$.

Motivation & Objective

• To design fast distributed algorithms for approximating maximum independent set and maximum weight matching in large-scale networks.
• To overcome the challenge of simulating algorithms on line graphs in the CONGEST model without incurring additional congestion overhead.
• To achieve provably optimal round complexity in terms of maximum degree Δ for weighted matching approximation.
• To provide both deterministic and randomized solutions with strong approximation guarantees and low communication cost.

Proposed method

• Adapts the local ratio technique to the distributed setting by iteratively computing maximal independent sets (MIS) and performing weight reductions in parallel.
• Groups nodes into log W layers based on weight, ensuring each iteration reduces the topmost layer’s weight, leading to O(MIS(G) · log W) round complexity.
• Uses (Δ+1)-coloring to deterministically simulate MIS-based weight reduction, achieving O(Δ + log* n) rounds without relying on randomized MIS algorithms.
• Employs a novel simulation mechanism for line graph algorithms that avoids the typical O(Δ) congestion penalty in CONGEST.
• Applies a randomized bipartite transformation and iterative matching to generalize results from bipartite to general unweighted graphs.
• Uses probabilistic analysis with Chernoff bounds to show high-probability approximation guarantees under random node partitioning and proposal strategies.

Experimental results

Research questions

• RQ1 Can a distributed Δ-approximation for maximum weight independent set be achieved with low round complexity in the CONGEST model?
• RQ2 How can line graph simulations be emulated in CONGEST without incurring a multiplicative O(Δ) overhead?
• RQ3 What is the optimal round complexity for (2+ε)-approximate maximum weight matching in the CONGEST model?
• RQ4 Can (1+ε)-approximation for unweighted matching be achieved in O(log Δ / log log Δ) rounds?
• RQ5 What is the impact of randomized node partitioning on achieving high-accuracy matching approximations in general graphs?

Key findings

• A deterministic Δ-approximation algorithm for maximum weight independent set runs in O(Δ + log* n) rounds using graph coloring.
• A randomized Δ-approximation for MaxIS completes in O(MIS(G) · log W) rounds, with MISE(G) bounded by O(log n) in practice.
• The algorithm achieves a 2-approximation for maximum weight matching without additional round cost by simulating line graph algorithms efficiently.
• For (2+ε)-approximate maximum weight matching, the algorithm runs in O(log Δ / log log Δ) rounds, matching the theoretical lower bound up to logarithmic factors.
• A (1+ε)-approximation for unweighted matching is achievable in O(log Δ / log log Δ) rounds, making it the first algorithm with provably optimal round complexity in terms of Δ.
• The simulation mechanism for line graph algorithms avoids O(Δ) congestion overhead, enabling efficient distributed matching approximation.