A General Framework for Computing the Nucleolus Via Dynamic Programming

This paper defines a general class of cooperative games for which the nucleolus is efficiently computable. This class includes new members for which the complexity of computing their nucleolus was not previously known. We show that when the minimum excess coalition problem of a cooperative game can be formulated as a hypergraph dynamic program its nucleolus is efficiently computable. This gives a general technique for designing efficient algorithms for computing the nucleolus of a cooperative game. This technique is inspired by a recent result of Pashkovich (2018) on weighted voting games. However our technique significantly extends beyond the capabilities of previous work. We demonstrate this by applying it to give an algorithm for computing the nucleolus of b-matching games in polynomial time on graphs of bounded treewidth.

💡 Research Summary

The paper introduces a broad algorithmic framework for efficiently computing the nucleolus—a central solution concept in cooperative game theory—by leveraging dynamic programming (DP) techniques on hypergraphs. The nucleolus is defined as the imputation that lexicographically maximizes the vector of excesses (the difference between the allocated payoff to any coalition and its value). Traditional approaches, notably the Maschler‑Peleg‑Shapley (MPS) scheme, solve a sequence of linear programs (LPs) whose constraints involve all possible coalitions, leading to an exponential number of constraints and requiring a sophisticated separation oracle.

The authors observe that the key subproblem in each MPS iteration is the “minimum excess coalition problem”: given an imputation x, find a coalition S that minimizes x(S) − ν(S). They propose to model this subproblem as a DP on a directed acyclic hypergraph (DAG). If such a DP can be constructed and solved in time T, then the nucleolus can be computed in polynomial time in T (Theorem 1). The crucial technical contribution is showing that adding congruence (modular) constraints to the DP—required for the MPS scheme’s refined LPs—only incurs a polynomial overhead (Theorem 3). To achieve this, they develop a new “linear subspace avoidance” problem and prove a series of lemmas that allow the DP to be modified so that its feasible solutions avoid a given linear subspace while preserving tractability. This extends the earlier result of Pashkovich (2018), who handled weighted voting games via a knapsack‑type DP, to a much wider class of combinatorial optimization games.

The framework is then instantiated for b‑matching games, a natural generalization of matching games where each vertex v may be incident to at most b(v) edges in the selected matching, and edges carry weights. Computing the nucleolus of general b‑matching games is known to be NP‑hard (e.g., when b(v) ≥ 3). The authors restrict attention to graphs of bounded treewidth. By employing a tree decomposition, they design a hypergraph DP that processes each bag of the decomposition, encoding the number of incident edges selected for each vertex and the partial matching structure. Lemma 12 shows that the minimum excess coalition can be found in time polynomial in the size of the graph when the treewidth is a fixed constant. Applying Theorem 1 yields Theorem 2: the nucleolus of weighted b‑matching games on bounded‑treewidth graphs can be computed in polynomial time.

Beyond b‑matching, the paper discusses how the same methodology applies to other combinatorial optimization games such as flow games, assignment games, and covering games, provided that the underlying optimization problem admits a DP formulation on a hypergraph. The authors also compare their results with prior hardness proofs (e.g., for weighted voting, spanning tree, and general b‑matching games) and highlight that their framework narrows the gap between tractable and intractable instances by identifying structural parameters (like treewidth) that render the nucleolus computation feasible.

In summary, the contributions are: (1) a general reduction from nucleolus computation to a minimum‑excess coalition problem; (2) a proof that hypergraph DP with congruence constraints remains polynomial‑time solvable; (3) a concrete polynomial‑time algorithm for the nucleolus of weighted b‑matching games on bounded‑treewidth graphs; and (4) a unifying perspective that extends the reach of dynamic‑programming‑based nucleolus algorithms far beyond the previously studied weighted voting games. This work advances both the theoretical understanding of nucleolus complexity and provides practical algorithmic tools for a wide range of cooperative games with compact representations.