In the Minimum Bisection problem, input is a graph $G$ and the goal is to partition the vertex set into two parts $A$ and $B$, such that $||A|-|B|| \le 1$ and the number $k$ of edges between $A$ and $B$ is minimized. This problem can be viewed as a clustering problem where edges represent similarity, and the task is to partition the vertices into two equally sized clusters, while minimizing the number of pairs of similar objects that end up in different clusters. In this paper, we initiate the study of a fair version of Minimum Bisection. In this problem, the vertices of the graph are colored using one of $c \ge 1$ colors. The goal is to find a bisection $(A, B)$ with at most $k$ edges between the parts, such that for each color $i\in [c]$, $A$ has exactly $r_i$ vertices of color $i$. We first show that Fair Bisection is $W$[1]-hard parameterized by $c$ even when $k = 0$. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class $i$ has {\em exactly} $r_i$ vertices in $A$. In particular, we give an $f(k,c,\epsilon)n^{O(1)}$ time algorithm that finds a balanced partition $(A, B)$ with at most $k$ edges between them, such that for each color $i\in [c]$, there are at most $(1\pm \epsilon)r_i$ vertices of color $i$ in $A$. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP '18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing '14]). Along the way, we also derive a combinatorial result regarding tree decompositions of graphs.

翻译：在最小二分问题中，输入是一个图 $G$，目标是将顶点集划分为两部分 $A$ 和 $B$，使得 $||A|-|B|| \le 1$ 且 $A$ 与 $B$ 之间的边数 $k$ 最小化。该问题可视为一个聚类问题，其中边表示相似性，任务是将顶点划分为两个大小相等的聚类，同时最小化最终位于不同聚类的相似对象对的数量。本文中，我们首次研究了最小二分的公平版本。在此问题中，图的顶点使用 $c \ge 1$ 种颜色之一着色。目标是找到一个二分 $(A, B)$，使得两部分之间至多有 $k$ 条边，且对于每种颜色 $i\in [c]$，$A$ 恰好有 $r_i$ 个颜色为 $i$ 的顶点。我们首先证明，即使当 $k = 0$ 时，Fair Bisection 关于参数 $c$ 是 $W$[1]-难的。另一方面，我们的主要技术贡献表明，这一困难结果仅仅源于每种颜色类 $i$ 在 $A$ 中恰好有 $r_i$ 个顶点这一非常严格的要求。特别地，我们给出一个运行时间为 $f(k,c,\epsilon)n^{O(1)}$ 的算法，能够找到一个平衡划分 $(A, B)$，使得两部分之间至多有 $k$ 条边，且对于每种颜色 $i\in [c]$，$A$ 中颜色为 $i$ 的顶点数至多为 $(1\pm \epsilon)r_i$。我们的近似算法最好被视为一个概念验证，表明由 [Lampis, ICALP '18] 引入的用于处理有界树宽或团宽问题的 FPT 近似算法技术，即使在无界宽度的图上也能被有效利用。关键洞察在于，Lampis 的技术适用于具有不可破袋（如 [Cygan 等人, SIAM Journal on Computing '14] 中引入）的树分解。在此过程中，我们还推导出关于图的树分解的一个组合结果。