发表机构
Faculty of Mathematics and Physics, University of Ljubljana; Institute of Mathematics, Physics and Mechanics; Abelium d.o.o.（卢布尔雅那大学数学物理学院; 数学、物理与力学研究所; 阿贝利乌姆有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明计算辫子移动图、高阶 Bruhat 序及定向拟阵突变图中的距离是 NP 困难的，通过从顶点覆盖归约构造实例，并推广到所有固定秩和余秩。
AI 中文摘要
最长置换 $w_0\in S_m$ 的约化字（在交换操作下等价）编码了 $m$ 条伪线的布线图，而辫子移动 $\sigma_i\sigma_{i+1}\sigma_i\leftrightarrow\sigma_{i+1}\sigma_i\sigma_{i+1}$ 翻转其一个三角形。我们证明，判定两个布线图是否在给定的辫子移动距离内是 NP 完全的。等价地，判定翻转距离在高阶 Bruhat 序 $B(m,2)$ 中、在伪线标记排列的三角形翻转中、以及在 $2m$ 边形的菱形平铺中都是 NP 完全的；这回答了关于伪线排列翻转图的一个开放问题（SODA 2024）。该归约来自顶点覆盖问题：对于每个图 $G$，我们构造两个布线图，其辫子移动距离为 $|D|+2\\,\mathrm{VC}(G)$，其中 $\mathrm{VC}(G)$ 是 $G$ 的顶点覆盖数，而汉明距离 $|D|$ 是三角形方向不同的线三元组的数量。该构造粘合了一个七线装置的副本，其辫子移动距离比其汉明距离大 $2$；在两个图中翻转两个不相交三角形中的任意一个即可消除该超出量。在无穷远处添加一条线将构造转移到秩为 $3$ 的均匀定向拟阵，其中两个定向拟阵在突变图中相邻当且仅当它们的 chirotope 在一个基上不同；因此，判定秩 $3$ 中的突变距离是 NP 完全的。Rambau 的 signotope 展开的迭代放大了超出量，并在每个固定秩 $r\ge 4$ 中给出 NP 困难性。通过对偶性，我们获得了每个固定余秩至少为 $3$ 的相应结果。相同的实例表明，高阶 Bruhat 序 $B(n,k)$ 中的翻转距离对于每个固定 $k\ge 2$ 是 NP 完全的。
英文摘要
A reduced word of the longest permutation $w_0\in S_m$, taken up to commutations, encodes a wiring diagram of $m$ pseudolines, and a braid move $σ_iσ_{i+1}σ_i\leftrightarrowσ_{i+1}σ_iσ_{i+1}$ flips one of its triangles. We prove that deciding whether two wiring diagrams are within a given braid-move distance is NP-complete. Equivalently, deciding flip distance is NP-complete in the higher Bruhat order $B(m,2)$, for triangle flips of marked arrangements of pseudolines, and for rhombic tilings of a $2m$-gon; this answers an open question on flip graphs of pseudoline arrangements (SODA 2024). The reduction is from Vertex Cover: for every graph $G$ we construct two wiring diagrams at braid-move distance $|D|+2\,\mathrm{VC}(G)$, where $\mathrm{VC}(G)$ is the vertex cover number of $G$ and the Hamming distance $|D|$ is the number of triples of wires whose triangles are oriented differently. The construction glues copies of a seven-wire gadget whose braid-move distance exceeds its Hamming distance by $2$; flipping either of two disjoint triangles in both diagrams removes the excess. Adding a line at infinity transfers the construction to rank-$3$ uniform oriented matroids, where two oriented matroids are adjacent in the mutation graph if their chirotopes differ in one basis; so deciding mutation distance is NP-complete in rank $3$. An iterate of Rambau's expansion of signotopes amplifies the excess and gives NP-hardness in every fixed rank $r\ge 4$. By duality we obtain the corresponding results for every fixed corank at least $3$. The same instances show that flip distance in the higher Bruhat orders $B(n,k)$ is NP-complete for every fixed $k\ge 2$.
Comments37 pages, 4 figures. Code and computational checks: https://github.com/tilenmarc/braid-bruhat-mutation-distance
