Multicut Problems in Almost-Planar Graphs: The Dependency of Complexity on the Demand Pattern

TL;DR

This work resolves how the complexity of Multicut on almost-planar graphs depends on the demand pattern and planarity-approximation parameters. By introducing and exploiting multicut duals and carefully controlling treewidth through states and reachability arguments, the authors give a complete fixed-parameter tractability picture when the demand graph class H has bounded distance to extended bicliques, with running time f(π,t)·n^{O(√μ)} where μ is that distance; they also extend the analysis to crossing-number parameters cr in both edge-weighted and unweighted settings, yielding a cohesive theory that ties planarity closeness (via π, g, and cr) to tractability and ETH-based lower bounds. The key technical innovations include a robust state-reduction framework (making states relevant and dealing with incomplete/complete states), a structural decomposition around the multicut dual with bounded treewidth after removing fixed-face regions, and a disciplined reduction to smaller instances that preserves optimality. Together, these results fully delineate how the proximity to planar graph structure alters Multicut’s exponent in the running time, and they unify planar, genus, and crossing-number regimes under a single algorithmic paradigm with explicit dependence on μ, π, t, and cr. The findings sharpen our understanding of when seemingly intractable instances become efficiently solvable and provide practical FPT techniques for a broad class of demand configurations in almost-planar graphs.

Abstract

Given a graph $G$, a set $T$ of terminal vertices, and a demand graph $H$ on $T$, the \textsc{Multicut} problem asks for a set of edges of minimum weight that separates the pairs of terminals specified by the edges of $H$. The \textsc{Multicut} problem can be solved in polynomial time if the number of terminals and the genus of the graph is bounded (Colin de Verdière [Algorithmica, 2017]). Focke et al.~[SoCG 2024] characterized which special cases of Multicut are fixed-parameter tractable parameterized by the number of terminals on planar graphs. Moreover, they precisely determined how the parameter genus influences the complexity and presented partial results of this form for graphs that can be made planar by the deletion of $π$ edges. We complete the picture on how this parameter $π$ influences the complexity of different special cases and precisely determine the influence of the crossing number. Formally, let $\mathcal{H}$ be any class of graphs (satisfying a mild closure property) and let Multicut$(\mathcal{H})$ be the special case when the demand graph $H$ is in $\mathcal{H}$. Our first main result is showing that if $\mathcal{H}$ has the combinatorial property of having bounded distance to extended bicliques, then Multicut$(\mathcal{H})$ on unweighted graphs is FPT parameterized by the number $t$ of terminals and $π$. For the case when $\mathcal{H}$ does not have this combinatorial property, Focke et al.~[SoCG 2024] showed that $O(\sqrt{t})$ is essentially the best possible exponent of the running time; together with our result, this gives a complete understanding of how the parameter $π$ influences complexity on unweighted graphs. Our second main result is giving an algorithm whose existence shows that the parameter crossing number behaves analogously if we consider weighted graphs.

Figures (1)

• Figure 1: The running time of the algorithms for Multicut($\mathcal{H}$) with the different parameterizations. The new results of the paper are marked by (*). Note that, for every fixed projection-closed class $\mathcal{H}$, the exponents of the running time are tight (up to logarithmic factors).

Theorems & Definitions (45)

• Theorem 1.1: focke_et_al:LIPIcs.SoCG.2024.57
• Theorem 1.2: focke2025journal
• Theorem 1.3: focke2025journal
• Theorem 1.4: focke_et_al:LIPIcs.SoCG.2024.57
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Proposition 2.2