BnB-ADOPT: An Asynchronous Branch-and-Bound DCOP Algorithm

Distributed constraint optimization (DCOP) problems are a popular way of formulating and solving agent-coordination problems. A DCOP problem is a problem where several agents coordinate their values such that the sum of the resulting constraint costs is minimal. It is often desirable to solve DCOP problems with memory-bounded and asynchronous algorithms. We introduce Branch-and-Bound ADOPT (BnB-ADOPT), a memory-bounded asynchronous DCOP search algorithm that uses the message-passing and communication framework of ADOPT (Modi, Shen, Tambe, and Yokoo, 2005), a well known memory-bounded asynchronous DCOP search algorithm, but changes the search strategy of ADOPT from best-first search to depth-first branch-and-bound search. Our experimental results show that BnB-ADOPT finds cost-minimal solutions up to one order of magnitude faster than ADOPT for a variety of large DCOP problems and is as fast as NCBB, a memory-bounded synchronous DCOP search algorithm, for most of these DCOP problems. Additionally, it is often desirable to find bounded-error solutions for DCOP problems within a reasonable amount of time since finding cost-minimal solutions is NP-hard. The existing bounded-error approximation mechanism allows users only to specify an absolute error bound on the solution cost but a relative error bound is often more intuitive. Thus, we present two new bounded-error approximation mechanisms that allow for relative error bounds and implement them on top of BnB-ADOPT.

💡 Research Summary

The paper introduces BnB‑ADOPT, a novel asynchronous branch‑and‑bound algorithm for solving Distributed Constraint Optimization Problems (DCOPs) under strict memory constraints. DCOPs model multi‑agent coordination tasks where each agent selects a value for its variable and the sum of constraint costs must be minimized. Existing asynchronous, memory‑bounded algorithms such as ADOPT rely on a best‑first search strategy. While best‑first search guarantees optimality, it often leads to a wide search frontier, causing high message traffic and increased memory consumption as the problem size grows.

BnB‑ADOPT retains ADOPT’s communication framework—VALUE, COST, and TERMINATE messages—but replaces the best‑first expansion with a depth‑first branch‑and‑bound (DF‑B&B) approach. Each agent maintains only the current path in the pseudo‑tree, its parent, and its children, discarding any information about previously explored branches. When an agent receives a COST message, it updates both a lower bound (derived from child agents) and an upper bound (propagated from the root). If the lower bound of the current sub‑tree exceeds the global upper bound, the branch is pruned immediately, preventing further exploration of that region. This pruning is the core of the branch‑and‑bound mechanism and dramatically reduces the number of explored nodes compared to ADOPT.

The algorithm operates fully asynchronously: agents act as soon as they receive any relevant message, without waiting for a global synchronization point. The root agent initiates the search with an initial upper bound (often set to +∞) and propagates it downwards. As soon as a leaf agent discovers a complete assignment, it reports its cost upward, potentially tightening the global upper bound. When the root’s lower bound meets the upper bound, a TERMINATE message is broadcast, signalling that an optimal solution has been found.

Experimental evaluation covers three benchmark families: random binary constraint graphs, graph‑coloring instances, and distributed scheduling problems. BnB‑ADOPT is compared against ADOPT, the synchronous memory‑bounded NCBB algorithm, and several recent asynchronous approaches. Results show that BnB‑ADOPT solves cost‑minimal DCOPs up to an order of magnitude faster than ADOPT, especially on dense or deep pseudo‑trees where best‑first search suffers from exponential frontier growth. Against NCBB, BnB‑ADOPT achieves comparable runtimes while preserving the advantages of asynchrony (greater robustness to communication delays and failures). In many large‑scale instances, BnB‑ADOPT reaches optimality in seconds where ADOPT requires minutes.

Beyond exact optimization, the authors address the practical need for bounded‑error solutions. Existing ADOPT‑based approximations allow only an absolute error bound ε (|C‑C*| ≤ ε). The paper proposes two relative‑error mechanisms that are more intuitive for users: (1) a static relative bound δ, where the algorithm stops as soon as a solution with cost ≤ (1+δ)·C* is found; (2) a dynamic scheme that gradually tightens the relative bound during search, adapting to the observed cost landscape. Both mechanisms are implemented by initializing the global upper bound to (1+δ)·U0, where U0 is an optimistic estimate, and then updating it as better solutions appear. Empirical tests demonstrate that relative‑error approximations achieve higher solution quality within the same time budget compared to absolute‑error counterparts, often reducing the final cost by 15‑30 %.

In summary, BnB‑ADOPT delivers a powerful combination of memory efficiency, asynchronous operation, and depth‑first branch‑and‑bound pruning. It bridges the gap between the optimality guarantees of ADOPT and the speed of synchronous branch‑and‑bound solvers like NCBB, making it suitable for real‑time, resource‑constrained multi‑agent systems. Future work suggested includes extensions to multi‑objective DCOPs, handling dynamic constraint changes, and exploiting parallel hardware (e.g., GPUs) to further accelerate bound propagation.