Computing distances in braid-move graphs, higher Bruhat orders, and oriented-matroid mutation graphs is NP-hard

Abstract

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 ${\sigma }_i{\sigma }_{i+1}{\sigma }_i\leftrightarrow {\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$ 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 $k\ge 2$.

37 pages, 4 figures. Code and computational checks: https://github.com/tilenmarc/braid-bruhat-mutation-distance