Distributed Greedy Algorithms

• Distributed Greedy Algorithms are approaches that make locally optimal choices across decentralized agents to approximate combinatorial optimization solutions.
• They are applied in network consensus, resource allocation, and sparsity-driven learning, leveraging local communications to accelerate convergence.
• The methods balance trade-offs in computation, memory, and communication, enabling scalable and resilient optimization in large-scale distributed systems.

A distributed greedy algorithm refers to any class of algorithms that addresses combinatorial optimization or inference by harnessing the greedy selection principle—making the locally optimal choice at each step—in a distributed computational or communication setting. These algorithms are foundational in decentralized consensus, networked resource allocation, multi-agent task assignment, submodular maximization, and large-scale machine learning, where the core challenge is to execute greedy logic across multiple agents, nodes, or processing units with limited coordination, often with stringent communication and memory constraints.

1. Greedy Algorithms in Distributed Consensus and Coordination

Distributed greedy algorithms have been extensively applied to average consensus and network resource allocation, exploiting only local state and neighbor exchanges. In the context of distributed averaging, the Greedy Gossip with Eavesdropping (GGE) algorithm (0909.1830) is a canonical example. Each iteration, a randomly selected node chooses, among its immediate neighbors, the one with the most discrepant value and performs a pairwise averaging update. GGE leverages the broadcast nature of wireless networks, permitting nodes to track neighborhood states via eavesdropping and thus facilitating greedy partner selection.

This approach departs from randomized gossip algorithms by prioritizing maximum local error reduction per communication, which empirically results in significantly faster convergence on a range of network topologies. Mathematically, each averaging step can be interpreted as an incremental subgradient method applied to a convex energy function measuring local discrepancies, and the consensus is achieved by successive greedy minimizations.

The distributed greedy framework also appears in maximal matching algorithms for anonymous, edge-colored graphs (Hirvonen et al., 2011), where the procedure of sequentially processing edges color-by-color in $k-1$ rounds is proven to be optimal in terms of round complexity.

2. Greedy Pursuit and Learning in Distributed Signal Processing

Distributed greedy pursuit algorithms play a central role in distributed compressed sensing and sparsity-constrained learning. Signals at distributed nodes are typically modeled as having partially common and partially private support (Sundman et al., 2013, Sundman et al., 2014). Each node operates a local greedy pursuit—such as Orthogonal Matching Pursuit (OMP), Subspace Pursuit (SP), or more advanced techniques like FROGS (Forward-Reverse OMP)—to estimate its signal’s sparse support. Instead of exchanging raw measurement vectors, nodes share only support-set estimates with neighbors, minimizing communication overhead.

Consensus on the joint support is achieved via voting or fusion mechanisms—nodes aggregate support estimates received via neighborhood communication and refine their own estimate accordingly. Theoretical analysis, often via the restricted isometry property (RIP), shows that convergence to the correct (joint) support set improves predictably with increased network connectivity and communication iterations. Recent analysis formalizes the probabilistic improvement brought by democratic voting: multiple independent support estimates are combined so that the chance of correctly identifying the joint support increases with the number of participating nodes (Sundman et al., 2014).

In distributed, sparsity-promoting learning (e.g., Distributed Hard Thresholding Pursuit and Greedy Diffusion LMS (Chouvardas et al., 2014)), greedy support identification and coefficient estimation are augmented by local neighborhood cooperation and recursive consensus averaging, enabling scalability and adaptivity under dynamic data arrival.

3. Distributed Submodular Maximization and Minimization

Distributed greedy algorithms provide provably good heuristics for maximizing (and, more recently, minimizing) submodular functions under cardinality, matroid, or partition constraints. In the maximization setting, agents sequentially or in parallel select elements that greedily maximize the marginal gain with respect to the system-level objective (Sun et al., 2020, Robey et al., 2019). When executed sequentially and with global state information, the greedy algorithm achieves a $(1-1/e)$-approximation for monotone submodular objectives, but this guarantee drops to $1/2$ under distributed partition matroid constraints and naive sequential execution.

Recent advances have overcome this barrier by "lifting" the problem to the continuous domain using the multilinear extension and running parallel, consensus-based message-passing algorithms (such as the Constraint-Distributed Continuous Greedy (CDCG) algorithm (Robey et al., 2019)), which recover the tight $(1-1/e)$ efficiency while maintaining distributed implementation via local gradient computation and update averaging.

For minimization, distributed greedy column generation approaches (Testa et al., 2018) reformulate the submodular minimization problem as a linear program over the base polyhedron, with the exponential pricing problem efficiently solved via a greedy routine at each agent. Through asynchronous message passing and local LP updates, finite-time convergence to the optimal solution is achieved even in unreliable and time-varying networks, as numerically validated for distributed $s$–$t$ minimum cut problems.

4. Parallelized and Memory-Constrained Distributed Greedy Algorithms

The scalability of greedy algorithms in distributed environments hinges on the ability to parallelize selection and aggregation steps without compromising approximation guarantees or overwhelming computational resources. The distributed greedy algorithm for column subset selection achieves this by leveraging randomized composable core-sets: random partitioning of data among machines, followed by local greedy selection and a global aggregation phase (Altschuler et al., 2016). Theoretical analysis establishes that such partitioning is essential for maintaining constant-factor approximation bounds even under severe distribution.

For very large-scale problems, "multi-level" distributed greedy algorithms such as GreedyML (Gopal et al., 15 Mar 2024) overcome memory bottlenecks by organizing machines in a hierarchical accumulation tree, recursively merging partial greedy solutions at each internal node. This approach avoids the centralization step that is prohibitive in single-level distributed algorithms like RandGreedi, thereby enabling large $k$-cover, $k$-dominating set, and $k$-medoid instances to be solved where prior algorithms fail. The approximation ratio deteriorates by at most a factor of $L+1$ (with $L$ accumulation levels), but empirical results show solution quality remains comparable to centralized or prior distributed greedy approaches, with substantial improvements in critical path runtime and maximum per-node memory requirements.

5. Trade-Offs, Limitations, and Communication Constraints

Distributed greedy algorithms must navigate fundamental trade-offs between performance, communication, memory usage, and execution time. For gossip-type algorithms (e.g., GGE (0909.1830), SGG (Shin et al., 2019)), increasing the scope of greedy selection (e.g., searching more neighbors) accelerates convergence but increases communication and state maintenance costs; adaptive strategies (Shin et al., 2019) enable tuning this trade-off.

Message passing order, network topology, and the structuring of communication steps have direct impact on the efficiency and guarantees achievable. For the distributed greedy submodular maximization framework, parallelization via partitioned agent execution introduces "blind spots" that degrade the competitive ratio: optimal batch partitioning strategies achieve the best possible bounds given the number of parallel iterations (Sun et al., 2020). Execution order in greedy multi-agent decision-making is shown to determine the communication complexity, with well-chosen (DFS-inspired) agent orderings enabling linear-time greedy execution, whereas poor orderings result in quadratic delays (Konda et al., 2021).

Moreover, in settings with limited information due to communication unreliability or restricted observability, the worst-case performance of distributed greedy methods degrades according to both curvature of the objective function and structural network measures such as the clique cover number (Biggs et al., 2022). Real-world experiments in robotic search planning validate these theoretical guarantees even when the underlying objectives lack submodularity.

6. Applications and Theoretical Implications

Distributed greedy algorithms are fundamental in a wide array of domains, including wireless sensor networks, distributed control, cooperative robotics, resource allocation, collective communications, large-scale clustering, and feature selection. Their ability to achieve provable approximation guarantees with limited per-node computation and memory, combined with robustness to unreliable networks, makes them central in decentralized optimization for modern large-scale systems.

The field continues to advance via improvements in approximation bounds, the development of more sophisticated parallel and hierarchical merging strategies, augmentation with consensus and voting mechanisms, and integration with privacy-preserving and robust planning strategies.

7. Future Directions and Challenges

Key open challenges include the development of distributed greedy algorithms for non-monotonic or non-submodular objectives with rigorous performance guarantees, further minimization of communication and memory constraints (including fully asynchronous models), adaptation to dynamic or mobile network topologies, and the incorporation of privacy-preserving and adversarial-robust mechanisms. Moreover, closing the remaining gaps between centralized and distributed achievable approximation factors—especially for matroid and combinatorial constraints beyond cardinality—remains a significant direction for both theory and practice. The ongoing integration of distributed greedy logic with continuous relaxation, subgradient frameworks, and local learning holds promise for scalable, resilient optimization in ever-larger and more heterogeneous multi-agent systems.