[Paper Review] A deterministic polynomial kernel for Odd Cycle Transversal and Vertex Multiway Cut in planar graphs

Bart M. P. Jansen, Marcin Pilipczuk+1 more|arXiv (Cornell University)|Oct 2, 2018

VLSI and FPGA Design Techniques4 citations

TL;DR

This paper presents the first deterministic polynomial kernel for Odd Cycle Transversal and Vertex Multiway Cut in planar graphs, parameterized by solution size. It introduces a novel sparsification technique that preserves essential low-cost substructures in planar graphs, enabling efficient kernelization without randomness, and establishes a connection to Vertex Planarization that may aid in resolving its open kernelization status.

ABSTRACT

We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014). It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Steiner trees after adding all edges between pairs of vertices that lie on a common face. We also show connections between Vertex Multiway Cut and the Vertex Planarization problem, where the existence of a polynomial kernel remains an important open problem.

Motivation & Objective

• To resolve Saurabh's open question on the existence of a deterministic polynomial kernel for Odd Cycle Transversal in planar graphs.
• To extend this result to the Vertex Multiway Cut problem in planar graphs.
• To develop a new sparsification routine that preserves non-Steiner-tree substructures relevant to cut problems in planar graphs.
• To establish connections between Vertex Multiway Cut and Vertex Planarization, potentially aiding in the long-standing open problem of a polynomial kernel for Vertex Planarization.
• To provide a framework for kernelization of vertex-deletion problems in planar graphs that avoids reliance on randomized representative sets in gammoids.

Proposed method

• Design a novel sparsification routine for planar graphs that preserves the existence of low-cost subgraphs which become supergraphs of Steiner trees after adding face-adjacent edges.
• Construct a planar embedding of the reduced graph by embedding components of the original graph into faces of a grid-like auxiliary structure (H) based on terminal degrees and neighbor orderings.
• Use a grid-based construction (H) of size O(k^2 * D) where D is the sum of terminal degrees, to simulate terminal connectivity and enforce planarity constraints.
• Prove equivalence between solutions to Planar Vertex Multiway Cut and Vertex Planarization by embedding components into faces of H and ensuring edge crossings are avoided via structural constraints.
• Leverage the Robertson-Seymour theorem and minor-closed properties to relate Vertex Multiway Cut to the broader Vertex Planarization problem.
• Apply a four-row and four-column selection argument in H to detect unavoidable minors (like H₀) in any solution that fails to separate terminals, proving correctness of the reduction.

Experimental results

Research questions

• RQ1 Can a deterministic polynomial kernel be achieved for Odd Cycle Transversal in planar graphs, overcoming the reliance on randomized representative sets in prior work?
• RQ2 Does the Vertex Multiway Cut problem in planar graphs admit a deterministic polynomial kernel, especially with a solution size parameter independent of the number of terminals?
• RQ3 Can a sparsification technique be designed for planar graphs that preserves non-Steiner-tree substructures critical for cut problems, rather than just Steiner trees?
• RQ4 What structural connections exist between Vertex Multiway Cut and Vertex Planarization that could lead to a polynomial kernel for the latter?
• RQ5 Can the kernel size bounds be improved beyond the current high-degree polynomial bounds, particularly by reducing the sparsification routine's complexity?

Key findings

• The paper constructs a deterministic polynomial kernel for Odd Cycle Transversal in planar graphs, resolving Saurabh's open question.
• It provides a deterministic polynomial kernel for Vertex Multiway Cut in planar graphs, with kernel size bounded by a polynomial in the solution size k, independent of the number of terminals.
• The authors introduce a new sparsification technique that preserves subgraphs which become Steiner tree supergraphs after adding face-adjacent edges, enabling deterministic kernelization.
• The construction establishes a tight equivalence between solutions to Planar Vertex Multiway Cut and Vertex Planarization, showing that a solution to one implies a solution to the other.
• The method avoids randomized algorithms by replacing representative set techniques with a geometric, planar-embedding-based reduction using a structured grid H.
• The kernel size is bounded by a high-degree polynomial in k, though the exact degree is not minimized, leaving room for future improvements.