We extend the list of games where the nucleolus is computable in polynomial time. Based on the classical MPS scheme, nucleolus computation can be reduced to the problem of finding a coalition with minimum excess that does not belong to a given linear subspace. We call this problem LSA-MinExcess, and show that it is equivalent to NZ-MinExcess: Given integral values per player, find a coalition with minimum excess whose player values do not sum up to $0$. Exploiting this representation, we prove that the nucleolus is computable in polynomial time for arboricity games, network strength games, and certain $b$-matching games. Along these lines, we show that for $b$-matching games with $b \leq 2$, LSA-MinExcess is polynomially equivalent to Shortest Non-Zero Cycle. Further, we prove that in general, linear subspace avoidance strictly increases the complexity of the minimum excess problem, even for monotone games. We still provide a reduction that trades linear subspace avoidance against an arbitrarily small approximation error. Finally, we show that the nucleolus is unstable in the following sense: A small change in the value function of the game can lead to a change in the nucleolus that is exponential in the number of players.

翻译：我们扩展了可在多项式时间内计算核仁的博弈列表。基于经典MPS方案，核仁计算可简化为寻找不属于给定线性子空间且具有最小超额联盟的问题。我们将此问题称为LSA-MinExcess，并证明其等价于NZ-MinExcess：给定每个玩家的整数值，寻找玩家值之和不为$0$且具有最小超额的联盟。利用这一表示，我们证明了树度博弈、网络强度博弈以及特定$b$-匹配博弈的核仁可在多项式时间内计算。沿着这些思路，我们表明对于$b \leq 2$的$b$-匹配博弈，LSA-MinExcess多项式等价于最短非零环问题。此外，我们证明一般情况下，即使对于单调博弈，线性子空间规避也会严格增加最小超额问题的复杂度。我们仍提供一种可将线性子空间规避转化为任意小近似误差的归约方法。最后，我们证明核仁在以下意义上是不稳定的：博弈价值函数的微小变化可能导致核仁发生随玩家数量呈指数级的变化。