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Near-Optimal Coalition Structures in Polynomial Time

Angshul Majumdar

TL;DR

The paper analyzes the classic coalition structure generation problem under a simple random sparse-synergy model and shows a rigorous anytime separation: sparse relaxations (e.g., OMP and L1-regularization) can find near-optimal coalition structures in polynomial time with high probability, while DP and MILP approaches generally require exponential time to reach similar quality. By formalizing three algorithm classes and a high-probability margin, it establishes that sparse methods rapidly identify true synergy templates, yielding substantial welfare gains long before exact methods converge. These results explain why approximate, sparse techniques can outperform exact algorithms in practice for large-scale CSG, and they underscore the value of convex and greedy approaches in cooperative game settings. The findings contribute to a theoretical understanding of when and why low-dimensional, sparse strategies dominate in anytime optimization for coalition formation.

Abstract

We study the classical coalition structure generation (CSG) problem and compare the anytime behavior of three algorithmic paradigms: dynamic programming (DP), MILP branch-and-bound, and sparse relaxations based on greedy or $l_1$-type methods. Under a simple random "sparse synergy" model for coalition values, we prove that sparse relaxations recover coalition structures whose welfare is arbitrarily close to optimal in polynomial time with high probability. In contrast, broad classes of DP and MILP algorithms require exponential time before attaining comparable solution quality. This establishes a rigorous probabilistic anytime separation in favor of sparse relaxations, even though exact methods remain ultimately optimal.

Near-Optimal Coalition Structures in Polynomial Time

TL;DR

The paper analyzes the classic coalition structure generation problem under a simple random sparse-synergy model and shows a rigorous anytime separation: sparse relaxations (e.g., OMP and L1-regularization) can find near-optimal coalition structures in polynomial time with high probability, while DP and MILP approaches generally require exponential time to reach similar quality. By formalizing three algorithm classes and a high-probability margin, it establishes that sparse methods rapidly identify true synergy templates, yielding substantial welfare gains long before exact methods converge. These results explain why approximate, sparse techniques can outperform exact algorithms in practice for large-scale CSG, and they underscore the value of convex and greedy approaches in cooperative game settings. The findings contribute to a theoretical understanding of when and why low-dimensional, sparse strategies dominate in anytime optimization for coalition formation.

Abstract

We study the classical coalition structure generation (CSG) problem and compare the anytime behavior of three algorithmic paradigms: dynamic programming (DP), MILP branch-and-bound, and sparse relaxations based on greedy or -type methods. Under a simple random "sparse synergy" model for coalition values, we prove that sparse relaxations recover coalition structures whose welfare is arbitrarily close to optimal in polynomial time with high probability. In contrast, broad classes of DP and MILP algorithms require exponential time before attaining comparable solution quality. This establishes a rigorous probabilistic anytime separation in favor of sparse relaxations, even though exact methods remain ultimately optimal.
Paper Structure (15 sections, 4 theorems, 17 equations)

This paper contains 15 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results: Anytime Separation
  4. Random value model
  5. Anytime behaviour
  6. Class $\mathcal{A}_{DP}$.
  7. Class $\mathcal{A}_{MILP}$.
  8. Class $\mathcal{A}_{sparse}$.
  9. High–probability margin and concentration
  10. Polynomial–time near–optimality of sparse relaxation
  11. Exponential lower bounds for dynamic programming
  12. Exponential lower bounds for MILP classes
  13. Anytime separation
  14. Discussion
  15. Conclusion

Key Result

Theorem 3.1

Assume the model eq:model and margin condition eq:margin. Then with probability at least $1-1/n$, every algorithm in $\mathcal{A}_{sparse}$ that in each iteration selects a coalition $S$ of maximal residual value among its current candidates identifies all $T_{j}$ in at most $k$ iterations, and the In particular, if $\gamma\ge 4\sigma\sqrt{\log(2n)}$, then $V(\widehat{\mathcal{P}})\ge (1-\varepsi

Theorems & Definitions (7)

  • Theorem 3.1: Sparse relaxation
  • proof
  • Theorem 3.2: DP lower bound
  • proof
  • Theorem 3.3: MILP lower bound
  • proof
  • Corollary 3.4: Anytime separation