A Near-Linear Approximation Scheme for Multicuts of Embedded Graphs With a Fixed Number of Terminals

For an undirected edge-weighted graph $G$ and a set $R$ of pairs vertices called terminals, multicut is edges such that removing these from disconnects each pair in $R$. We provide algorithm computing $(1+\varepsilon)$-approximation the minimum time $(g+t)^{(O(g+t)^3)}\cdot(1/\varepsilon)^{O(g+t)} \cdot n \log n$, where $g$ genus $t$ number terminals. This tight several aspects, as problem both APX-hard W[1]-hard (parameterized by terminals), even on planar graphs (equivalently, when $g=0$). Our result, field fixed-parameter approximation algorithms, mostly relies concepts borrowed computational topology surfaces. In particular, we use extend various recent techniques concerning homotopy, homology, covering spaces. Interestingly, topological seem necessary for case. also exploit classical ideas stemming schemes low-dimensional geometric inputs. A key insight toward our result novel characterization union some Steiner trees universal cover surface which embedded.