For a digraph $D$ and some $X \subseteq V(D)$, the inversion of $X$ is the operation of flipping all arcs both of whose endvertices are in $X$. We initiate the study of establishing arc-connectivity properties by applying inversions of bounded or fixed size. For fixed-size inversions, the feasibility problem is interesting. For all integers $p \geq 2$ and $k \geq 1$, we give a characterization of the digraphs that can be made $k$-arc-strong by applying inversions of size exactly $p$, provided they are sufficiently large. For bounded-size inversions, the feasibility problem is easy, so we focus on minimising the number of inversions. We prove that for all integers $p\geq 3$ and $k \geq 1$ and any $ε>0$, there exists a polynomial-time $(4k-2+ε)$-approximation algorithm for computing the minimum number of inversions of size at most $p$ that make a given digraph $k$-arc-strong. This is in stark contrast to other results on inversion optimization problems. On the other hand, we show that for any $p\geq 3$ and $k \geq 1$ the problem is NP-hard, and, moreover, APX-hard. As a result on parameterized complexity, we show that for any $k \geq 2$, it is $W[1]$-hard with respect to $p$ to decide whether a given digraph can be made $k$-arc-strong by applying a single inversion of size at most $p$. We also prove that for a given multidigraph, it is $W[1]$-hard with respect to $\ell$ to decide whether it can be made 2-arc-strong by applying $\ell$ inversions of size 2.

翻译：对于有向图$D$和$X \subseteq V(D)$，$X$的反演是指翻转所有两个端点均在$X$中的弧的操作。我们通过应用有界或固定大小的反演，开启了建立弧连通性质的研究。对于固定大小的反演，可行性问题具有趣味性。对所有整数$p \geq 2$和$k \geq 1$，我们给出了通过精确大小为$p$的反演（在充分大的条件下）能使有向图成为$k$-弧强连通图的特征刻画。对于有界大小的反演，可行性问题较为简单，因此我们专注于最小化反演次数。我们证明：对所有整数$p\geq 3$、$k \geq 1$及任意$ε>0$，存在多项式时间的$(4k-2+ε)$-近似算法，用于计算使给定有向图$k$-弧强连通所需的大小至多为$p$的反演的最小次数。这与反演优化问题的其他结果形成鲜明对比。另一方面，我们证明：对任意$p\geq 3$和$k \geq 1$，该问题是NP难的，且进一步为APX难的。作为参数化复杂性的结果，我们证明：对于任意$k \geq 2$，通过应用单个大小至多为$p$的反演判断给定有向图能否成为$k$-弧强连通的问题关于参数$p$是$W[1]$-难的。同时，我们证明：对于给定的多重有向图，通过应用$\ell$个大小为2的反演判断其能否成为2-弧强连通的问题关于参数$\ell$是$W[1]$-难的。