Planar Multiway Cut with Terminals on Few Faces

2025-06-29

AI Summary

This paper studies the Edge Multiway Cut problem on planar graphs where all terminals lie on at most $k$ faces, parameterized by $k$. Prior best algorithms ran in $n^{O(sqrt{t})}$ time ($t$: number of terminals), while $k leq t$ and can be significantly smaller. We present the first $n^{O(sqrt{k})}$-time algorithm, introducing homotopy-theoretic techniques and sphere-cut decompositions to this problem. Our approach integrates global treewidth-based dynamic programming with Dreyfus–Wagner-style local optimization over face-enclosing subgraphs. This result matches the optimal parameter dependence for planar Steiner Tree and is tight under the Exponential Time Hypothesis (ETH)—no $n^{o(sqrt{k})}$ algorithm exists. The key innovation lies in exploiting the topological structure of planar embeddings to achieve exponential speedup driven by face count rather than terminal count, establishing a new paradigm for cut problems under geometric constraints.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We consider the extsc{Edge Multiway Cut} problem on planar graphs. It is known that this can be solved in $n^{O(sqrt{t})}$ time [Klein, Marx, ICALP 2012] and not in $n^{o(sqrt{t})}$ time under the Exponential Time Hypothesis [Marx, ICALP 2012], where $t$ is the number of terminals. A stronger parameter is the number $k$ of faces of the planar graph that jointly cover all terminals. For the related {sc Steiner Tree} problem, an $n^{O(sqrt{k})}$ time algorithm was recently shown [Kisfaludi-Bak et al., SODA 2019]. By a completely different approach, we prove in this paper that extsc{Edge Multiway Cut} can be solved in $n^{O(sqrt{k})}$ time as well. Our approach employs several major concepts on planar graphs, including homotopy and sphere-cut decomposition. We also mix a global treewidth dynamic program with a Dreyfus-Wagner style dynamic program to locally deal with large numbers of terminals.

Problem

• Solving Edge Multiway Cut in planar graphs efficiently
• Reducing complexity using terminal-covered face count
• Combining treewidth and Dreyfus-Wagner dynamic programming techniques

Innovation

• Uses homotopy and sphere-cut decomposition
• Combines global treewidth dynamic program
• Mixes Dreyfus-Wagner style dynamic program