  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.

Data Structures and Algorithms