Faster Optimal Coalition Structure Generation via Offline Coalition Selection and Graph-Based Search

TL;DR

The paper tackles the computationally intense Coalition Structure Generation problem by introducing SMART, a hybrid algorithm that blends offline coalition-size optimization with three complementary techniques: Complementarity-Based Dynamic Programming (CDP), Gradual Search with Dynamic Programming (GRAD), and Distributed Integer Partition Graph Search (DIPS). CDP provides a worst-case-optimized DP backbone by selecting two coalition-size sets offline and solving via two parallel DP processes; GRAD adaptively explores subspaces in parallel with offline SOFT-driven size-set selection and online pruning using upper bounds; DIPS distributes work across processes and uses branch-and-bound to prune unpromising subspaces. The authors prove completeness and analyze complexities, showing SSD scales as $\mathcal{O}(2^{2n} \times n^{2} \times \frac{e^{\pi\sqrt{2n/3}}}{n})$ while the overall SMART runtime remains bounded by $\mathcal{O}(3^{n})$, outperforming prior hybrids such as ODP-IP and BOSS. Empirical results across nine distributions and multiple agent counts demonstrate that SMART is consistently faster and more stable, achieving orders-of-magnitude speedups in generating optimal coalition structures. The work advances exact CSG by integrating offline coalition selection, dynamic programming refinements, and distributed graph search to deliver practical, scalable solutions for multi-agent systems.

Abstract

Coalition formation is a key capability in multi-agent systems. An important problem in coalition formation is coalition structure generation: partitioning agents into coalitions to optimize the social welfare. This is a challenging problem that has been the subject of active research for the past three decades. In this paper, we present a novel algorithm, SMART, for the problem based on a hybridization of three innovative techniques. Two of these techniques are based on dynamic programming, where we show a powerful connection between the coalitions selected for evaluation and the performance of the algorithms. These algorithms use offline phases to optimize the choice of coalitions to evaluate. The third one uses branch-and-bound and integer partition graph search to explore the solution space. Our techniques bring a new way of approaching the problem and a new level of precision to the field. In experiments over several common value distributions, we show that the hybridization of these techniques in SMART is faster than the fastest prior algorithms (ODP-IP, BOSS) in generating optimal solutions across all the value distributions.

Figures (10)

• Figure 1: A four-agent integer partition graph.
• Figure 2: Illustration of CDP on a 10-agent integer partition graph. CDP evaluates the coalitions of size $s$$\in \mathcal{BS}_{1}$=$\{2,4,6,10\}$ (Figure \ref{['CDP']}.a) and those of size $s$$\in \mathcal{BS}_{2}$=$\{2,8,10\}$ (Figure \ref{['CDP']}.b) in parallel. With the set $\mathcal{BS}_{1}$ (resp. $\mathcal{BS}_{2}$), CDP explores all the subspaces in Figure \ref{['CDP']}.a (resp. Figure \ref{['CDP']}.b), except the red ones. Nevertheless, CDP covers all the subspaces using $\mathcal{BS}_{1}$ and $\mathcal{BS}_{2}$. Hence, no node is missed by both sets, that is, no node is red in both figures.
• Figure 3: Illustration of the subspace distribution technique. The subspaces are sorted according to their upper bounds. The subspace [2,4,4] is the highest upper bound node and [1,4,5] is the lowest upper bound node. First, DIPS starts by searching the highest upper bound subspaces. Then, each time a GRAD or CDP process is released, DIPS uses the process to expand the parallelism of its search. For example in this Figure, when the released "GRAD process 1" is used by DIPS to search the node [1,1,1,3,4], the "DIPS process" searches another node.
• Figure 4: Run time of SMART, BOSS, ODP-IP, and POI.
• Figure 5: Coalition structure graph of 4 agents.

Theorems & Definitions (8)

• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof