Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern

TL;DR

The paper studies Multicut on graphs embedded in surfaces, focusing on how the demand pattern class $\mathcal{H}$ influences the complexity beyond the genus $g$ and terminal count $t$. It develops a framework based on dual multicut duals, pathwidth/treewidth bounds, and extended biclique decompositions to derive algorithmic upper bounds and ETH-based lower bounds, showing a dichotomy depending on $\mathcal{H}$'s structure. The main results establish polynomial-time solvability for trivial-demand-pattern classes, $f(t,g)n^{O(g)}$-time algorithms for classes with bounded distance to extended bicliques, and tight (up to logarithmic factors) lower bounds otherwise; they also resolve the $t=3$ Multiway Cut case and extend insights to Group 3-Terminal Cut. Collectively, the work clarifies when the complexity of Multicut is driven by the demand pattern versus the ambient genus, guiding efficient algorithms for structured demand graphs and delineating the boundaries of hardness in the fixed-genus regime.

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 $T\subseteq V(G)$ 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 $G\setminus C$. Colin de Verdière [Algorithmica, 2017] showed that Multicut with $t$ terminals on a graph $G$ of genus $g$ can be solved in time $f(t,g)n^{O(\sqrt{g^2+gt+t})}$. 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$(\mathcal{H})$.

Figures (4)

• Figure 1: Reducing a Multicut instance $(G',H')$ to $(G,H)$ where $H'$ is a projection of $H$ that is obtained by identifying the vertices $u$ and $v$ into $w$, and deleting $t_4$ and $t_5$. The corresponding modifications of $G$ are also illustrated. The edge $uv$ is bold to indicate that it has a (sufficiently) large weight.
• Figure 2: An example of a multicut dual. The yellow squares represent the terminals. The blue dashed curves represent the edges of $H$. The cyan graph represents a multicut dual.
• Figure 3: An example of an extended biclique partition.
• Figure 4: The black graph illustrates a minimal subcubic multicut dual $C$ for an instance $(G,H)$ where the vertices of $H$ are depicted in different shapes and colors according to their role in the extended biclique decomposition. From the minimality of $C$, it follows that, for two adjacent faces, at least one of them must contain a terminal from $X$, or otherwise one face contains a terminal from $B_1$ and the other a terminal from $B_2$. As a consequence, every degree-$3$ vertex of $C$ is incident to a face containing an element of $X$. Note that this is not necessarily true for degree-$2$ vertices, for instance consider the vertices of the inner cycle of the two components at the top of the drawing.

Theorems & Definitions (16)

• Theorem 1.1: Colin de Verdière ECDV
• Theorem 1.2: Cohen-Addad et al. Cohen-AddadVMM21
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 1.8
• Theorem 1.9