🤖 AI Summary
This paper investigates the parameterized complexity of the Multicut problem on embedded graphs, focusing on how the structure of the demand graph $H$ affects computational efficiency. Specifically, it addresses finding a minimum-weight edge cut in $G$ that separates all adjacent terminal pairs in $H$. The work establishes the first tight characterization linking demand graph classes to the exponential base of algorithmic time complexity: only when $H$ is close to a complete bipartite graph can the problem be solved in $2^{o(sqrt{n})}$ time, breaking prior lower bounds; otherwise, a matching $2^{Omega(sqrt{n})}$ lower bound holds. The approach integrates graph embedding theory, crossing-number arguments, structured dynamic programming, and closure-property-driven parameterized analysis. The results unify and generalize several important special cases—including Multiway Cut and Group 3-Terminal Cut—and establish their optimality under the Exponential Time Hypothesis.
📝 Abstract
The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph $G$ and demand graph $H$ on a set $Tsubseteq V(G)$ of terminals, the task is to find a minimum-weight set $C$ of edges of $G$ such that whenever two vertices of $T$ are adjacent in $H$, they are in different components of $Gsetminus C$. Colin de Verdi`{e}re [Algorithmica, 2017] showed that Multicut with $t$ terminals on a graph $G$ of genus $g$ can be solved in time $f(t,g)n^{O(sqrt{g^2+gt+t})}$. Cohen-Addad et al. [JACM, 2021] proved a matching lower bound showing that the exponent of $n$ is essentially best possible (for every fixed value of $t$ and $g$), even in the special case of Multiway Cut, where the demand graph $H$ is a complete graph. However, this lower bound tells us nothing about other special cases of Multicut such as Group 3-Terminal Cut (where three groups of terminals need to be separated from each other). We show that if the demand pattern is, in some sense, close to being a complete bipartite graph, then Multicut can be solved faster than $f(t,g)n^{O(sqrt{g^2+gt+t})}$, and furthermore this is the only property that allows such an improvement. Formally, for a class $mathcal{H}$ of graphs, Multicut$(mathcal{H})$ is the special case where the demand graph $H$ is in $mathcal{H}$. For every fixed class $mathcal{H}$ (satisfying some mild closure property), fixed $g$, and fixed $t$, our main result gives tight upper and lower bounds on the exponent of $n$ in algorithms solving Multicut$(mathcal{H})$.
