Balanced Stable Marriage (BSM) is a central optimization version of the classic Stable Marriage (SM) problem. We study BSM from the viewpoint of Parameterized Complexity. Informally, the input of BSM consists of n men, n women, and an integer k. Each person a has a (sub)set of acceptable partners, $\mathcal{A}(a)$, whom a ranks strictly; we use $p_a(b)$ to denote the position of $b\in \mathcal{A}(a)$ in a's preference list. The objective is to decide whether there exists a stable matching μ such that $\mathsf{balance}(\mu )\triangleq \max \{{\sum }_{(m,w)\in \mu }{p}_m(w),{\sum }_{(m,w)\in \mu }{p}_w(m)\}\le k$. In SM, all stable matchings match the same set of agents, $A^{\star }$ which can be computed in polynomial time. As $\mathsf{balance}(\mu )\ge \frac{|A^{\star }|}{2}$ for any stable matching μ, BSM is trivially fixed-parameter tractable (FPT) with respect to k. Thus, a natural question is whether BSM is FPT with respect to $k-\frac{|A^{\star }|}{2}$. With this viewpoint in mind, we draw a line between tractability and intractability in relation to the target value. This line separates additional natural parameterizations higher/lower than ours (e.g., we automatically resolve the parameterization $k-\frac{|A^{\star }|}{2}$). The two extreme stable matchings are the man-optimal ${\mu }_M$ and the woman-optimal ${\mu }_W$. Let $O_M={\sum }_{(m,w)\in {\mu }_M}{p}_m(w)$, and $O_W={\sum }_{(m,w)\in {\mu }_W}{p}_w(m)$. In this work, we prove that

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BSM parameterized by $t=k-\min \{O_M,O_W\}$ admits (1) a kernel where the number of people is linear in t, and (2) a parameterized algorithm whose running time is single exponential in t.

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BSM parameterized by $t=k-\max \{O_M,O_W\}$ is W[1]-hard.

平衡稳定婚姻 (BSM)是经典稳定婚姻 (SM)问题的中心优化版本。我们从参数化复杂性的角度研究BSM。非正式地，BSM的输入由n 个男人、n 个女人和一个整数k 组成。每个人a都有一组（子）可接受的合作伙伴，$\mathcal{一种}(\mathrm{一种})$，其中一个严格的行列; 我们用${磷}_{\mathrm{一种}}(乙)$ 表示位置 $乙\in \mathcal{一种}(\mathrm{一种})$在a的偏好列表中。目标是确定是否存在稳定匹配μ使得$\mathsf{平衡}(\mu )\triangleq \mathrm{最大限度}\{{\sum }_{(米,瓦)\in \mu }{磷}_{米}(瓦),{\sum }_{(米,瓦)\in \mu }{磷}_{瓦}(米)\}\le 克$. 在SM 中，所有稳定匹配都匹配同一组代理，${\mathrm{一种}}^{\star }$可以在多项式时间内计算。作为$\mathsf{平衡}(\mu )\ge \frac{|{\mathrm{一种}}^{\star }|}{2}$对于任何稳定匹配μ，BSM是相对于k 的平凡固定参数易处理 (FPT) 。因此，一个自然的问题是BSM是否是 FPT$克-\frac{|{\mathrm{一种}}^{\star }|}{2}$. 考虑到这一观点，我们在与目标值相关的易处理性和难处理性之间划清界限。这条线分隔了比我们更高/更低的额外自然参数化（例如，我们自动解决参数化$克-\frac{|{\mathrm{一种}}^{\star }|}{2}$）。两个极端稳定的匹配是人为最优的${\mu }_{米}$ 和女性最佳 ${\mu }_{宽}$. 让${哦}_{米}={\sum }_{(米,瓦)\in {\mu }_{米}}{磷}_{米}(瓦)$， 和 ${哦}_{宽}={\sum }_{(米,瓦)\in {\mu }_{宽}}{磷}_{瓦}(米)$. 在这项工作中，我们证明

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BSM参数化为$吨=克-\mathrm{分钟}\{{哦}_{米},{哦}_{宽}\}$承认 (1) 一个内核，其中人数在t 中是线性的，以及 (2) 一个参数化算法，其运行时间是t 中的单指数。

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BSM参数化为$吨=克-\mathrm{最大限度}\{{哦}_{米},{哦}_{宽}\}$ 是 W[1]-hard。