[Paper Review] GRASP for the Coalition Structure Formation Problem

Nicola Di Mauro, Teresa M. A. Basile+2 more|arXiv (Cornell University)|Apr 16, 2010

Game Theory and Voting Systems9 references3 citations

TL;DR

This paper proposes GRASP-CSF, a greedy adaptive search procedure that efficiently finds near-optimal coalition structures in multi-agent systems. By combining constructive and local search phases, it achieves near-optimal solutions up to 1,000x faster than exact algorithms like IDP, particularly excelling in time-constrained environments with high-dimensional problems.

ABSTRACT

The coalition structure formation problem represents an active research area in multi-agent systems. A coalition structure is defined as a partition of the agents involved in a system into disjoint coalitions. The problem of finding the optimal coalition structure is NP-complete. In order to find the optimal solution in a combinatorial optimization problem it is theoretically possible to enumerate the solutions and evaluate each. But this approach is infeasible since the number of solutions often grows exponentially with the size of the problem. In this paper we present a greedy adaptive search procedure (GRASP) to efficiently search the space of coalition structures in order to find an optimal one. Experiments and comparisons to other algorithms prove the validity of the proposed method in solving this hard combinatorial problem.

Motivation & Objective

• To address the NP-complete coalition structure formation (CSF) problem in multi-agent systems where exhaustive search is infeasible due to exponential growth in solution space.
• To develop a fast, anytime algorithm capable of returning high-quality solutions under time constraints, crucial for reactive agent systems.
• To outperform existing exact and heuristic algorithms in both solution quality and computation time for large-scale CSF problems.
• To evaluate the effectiveness of GRASP as a stochastic local search method for combinatorial optimization in coalition formation.

Proposed method

• GRASP-CSF employs a two-phase approach: a constructive phase using a greedy randomized adaptive procedure to build initial coalition structures, followed by a local search phase to improve solutions.
• The algorithm uses a randomized greedy selection strategy to prioritize high-value coalitions during construction, reducing bias toward suboptimal partitions.
• Local search employs neighborhood operators such as split, merge, extract, and exchange to explore adjacent coalition structures and escape local optima.
• The algorithm is configured with a maximum iteration limit (e.g., 20) to ensure termination and guarantee optimal solution recovery when needed.
• Solution quality is measured via relative quality: V(C)/V(C*) where C* is the optimal coalition structure.
• Experiments use uniform and non-uniform value distributions across 10 problem instances per agent count (14–20 agents), with time and quality averaged over runs.

Experimental results

Research questions

• RQ1 Can GRASP be effectively applied as an anytime algorithm for the coalition structure formation problem, delivering progressively better solutions under time limits?
• RQ2 Can GRASP-CSF find optimal coalition structures faster than state-of-the-art exact algorithms like IDP and IP?
• RQ3 How does GRASP-CSF compare in solution quality and speed to other metaheuristics such as Simulated Annealing and Sandholm et al.'s algorithm?
• RQ4 To what extent does the algorithm’s performance scale with increasing numbers of agents?

Key findings

• GRASP-CSF found optimal solutions in under 0.05 seconds on average, while IDP required up to 47.8 seconds, making GRASP-CSF up to 1,000 times faster for 20-agent problems.
• For 20 agents, GRASP-CSF achieved optimal solutions in just 0.0499 seconds on average, compared to 47.8 seconds for IDP.
• The algorithm reached near-optimal solutions (relative quality > 0.99) within the first 3 iterations, with average solution quality exceeding 0.995 after 10 iterations.
• GRASP-CSF consistently outperformed both Simulated Annealing and Sandholm et al.'s algorithm in solution quality, with lower relative error across all tested problem sizes (14–20 agents).
• The performance gap between GRASP-CSF and IDP widened with problem size, indicating superior scalability of the proposed method.
• Even with a fixed maxiter of 20, GRASP-CSF achieved 100% optimal solution recovery for 18–20 agents in all 10 instances, demonstrating robustness and reliability.