Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern
The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph G and a demand graph H on a set $$T\subseteq V(G)$$ T ⊆ 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 , th...
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| Vydáno v: | Discrete & computational geometry |
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| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
22.10.2025
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| ISSN: | 0179-5376, 1432-0444 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph G and a demand graph H on a set $$T\subseteq V(G)$$ T ⊆ 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 $$G\setminus C$$ G \ C . Colin de Verdière [ Algorithmica, 2017] showed that Multicut with t terminals on a graph G of genus g can be solved in time $$f(t,g)n^{O(\sqrt{g^2+gt+t})}$$ f ( t , g ) n O ( g 2 + g t + 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 $$f(t,g)n^{O(\sqrt{g^2+gt+t})}$$ f ( t , g ) n O ( g 2 + g t + t ) , and furthermore this is the only property that allows such an improvement. Formally, for a class $$\mathcal {H}$$ H of graphs, $$\textsc {Multicut}(\mathcal {H})$$ M U L T I C U T ( H ) is the special case where the demand graph H is in $$\mathcal {H}$$ H . For every fixed class $$\mathcal {H}$$ 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 $$\textsc {Multicut}(\mathcal {H})$$ M U L T I C U T ( H ) . |
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| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-025-00782-x |