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arXiv · 2610.29731

Computing the Nucleon of Matching Games

Haris Aziz·Gergely Csáji
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Abstract

The nucleon is the multiplicative counterpart of the nucleolus: it lexicographically maximizes the fraction of its stand-alone value that each coalition receives. We study exact nucleon computation in matching games and in two extensions: $b$-matching games, in which players may have several partners, and partitioned matching games, in which one player may own several vertices. We resolve three open problems from a recent survey. For ordinary matching games with nonnegative rational weights, we give a polynomial-time combinatorial algorithm using maximum-weight matching, shortest paths, and minimum-cost flow, ; it returns both a nucleon allocation and a compact description of the full nucleon. In contrast, computing any nucleon allocation is NP-hard for unit-weight $b$-matching games with capacities at most three and for unit-weight partitioned matching games of width at most three. Finally, for the unresolved capacity-two case, we show that exact nucleon computation is polynomial-time Turing reducible to SHORTEST NON-ZERO CYCLE.

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Haris Aziz, Gergely Csáji. 2026-10-21. Computing the Nucleon of Matching Games. https://arxiv.org/abs/2610.29731

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