发表机构
Czech Technical University, Prague（捷克布拉格工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究最小成本生成树博弈的核心非成员判定问题，证明其在接近平面图中仍为coNP难，并给出以支持规模、树宽等为参数的FPT算法及多种核化结果。
AI 中文摘要
最小成本生成树博弈（MSTG）是一种在无向边加权图$(G,w)$上进行的合作博弈，其中每个顶点对应一个玩家，每条边具有关联成本$w$。一个特殊的顶点$s \in V(G)$代表供应点或源点。对于任意玩家联盟$S$，特征成本函数$c(S)$定义为连接恰好$S \cup \{s\}$中顶点的、关于$w$的最小生成树的成本。本文研究判定MSTG核心成员资格的计算复杂性。一般而言，判定给定分配是否属于核心是\textsf{coNP}-困难的（Faigle等人，国际博弈论杂志，1997）。我们将核心识别问题命名为{\sc MSTG Core Non-Membership}。我们将困难性扩展到非常接近平面图的图类。在正面结果方面，我们在参数化复杂性框架内提出了若干算法结果。我们证明当以分配的支持规模为参数时，{\sc MSTG Core Non-Membership}是固定参数可处理的（FPT）。转向图的结构参数，我们证明该问题在树宽和符号邻域多样性参数下具有FPT算法。最后但同样重要的是，我们研究核化。虽然在一般图中，在标准复杂性理论假设下，{\sc MSTG Core Non-Membership}不存在以顶点覆盖数为参数的多项式核，但我们在平面图中设计了一个三次核。此外，在一般图中，我们针对符号邻域多样性获得二次核，针对参数反馈边数获得线性核。
英文摘要
Minimum-cost spanning tree game (MSTG) is a cooperative game played on an undirected edge-weighted graph $(G,w)$ representing the network, where each vertex corresponds to a player and each edge has an associated cost~$w$. A distinguished vertex $s \in V(G)$ represents the supply or source. For any coalition of players $S$, the characteristic cost function $c(S)$ is defined as the minimum cost of a spanning tree with respect to $w$, connecting exactly the vertices in $S \cup \{s\}$. In this paper we study the computational complexity of deciding core membership for MSTG. In general, deciding whether a given allocation is in the core is \textsf{coNP}-hard~(Faigle et al.,International Journal of Game Theory,1997). We study the core recognition problem under the name {\sc MSTG Core Non-Membership}. We extend the hardness to graphs which are very close to being planar. On the positive side, we present several algorithmic results within the framework of parameterized complexity. We show that {\sc MSTG Core Non-Membership} is fixed-parameter tractable when parameterized by the support size of the allocation. Turning into structural parameters of graphs, we show that the problem admits an FPT algorithm parameterized by treewidth and signed neighborhood diversity. Last but not least, we investigate kernelization. While in general graphs, under standard complexity-theoretical assumptions, {\sc MSTG Core Non-Membership} does not admit a polynomial kernel parameterized by the vertex cover number, we design a cubic kernel in planar graphs. Furthermore, in general graphs, we obtain quadratic kernel for signed neighborhood diversity and linear kernel for the parameter feedback edge number.
