[Paper Review] A Polynomial Kernel For Multicut In Trees

Nicolás Bousquet, Jean Daligault+2 more|ArXiv.org|Feb 6, 2009

Graph theory and applications4 citations

TL;DR

This paper presents a polynomial kernel of size $O(k^6)$ for the Multicut in Trees problem, resolving an open question on whether the problem admits a polynomial kernel. The authors design a set of safe reduction rules—based on request disjointness, directionality, inclusion, common factors, and dominating wingspans—that iteratively simplify the instance while preserving the solution, ensuring the kernel size is polynomial in $k$. The result confirms that Multicut in Trees is fixed-parameter tractable with a polynomial kernel, improving upon the previously known exponential kernel.

ABSTRACT

The MULTICUT IN TREES problem consists in deciding, given a tree, a set of requests (i.e. paths in the tree) and an integer k, whether there exists a set of k edges cutting all the requests. This problem was shown to be FPT by Guo and Niedermeyer. They also provided an exponential kernel. They asked whether this problem has a polynomial kernel. This question was also raised by Fellows. We show that MULTICUT IN TREES has a polynomial kernel.

Motivation & Objective

• To resolve the open question of whether Multicut in Trees admits a polynomial kernel, posed by Guo and Niedermeier and later by Fellows.
• To design a set of safe reduction rules that preserve the solution while reducing the instance size to polynomial in the parameter $k$.
• To extend kernelization techniques from caterpillar trees to general trees, using structural properties like wingspans and internal path dominance.
• To provide a constructive kernelization algorithm that runs in polynomial time and ensures the reduced instance size is bounded by a polynomial in $k$.

Proposed method

• Applying a sequence of six safe reduction rules in a fixed order: Unit Request, Disjoint Requests, Unique Direction, Inclusion, Common Factor, and Dominating Wingspan.
• Using the concept of wingspans—paths between quasi-R-neighbors—to identify and contract edges associated with $L_2$-leaves that dominate many endpoint-disjoint requests.
• Defining dominance relations between requests based on their internal paths: a request $R_1$ dominates $R_2$ if the internal path of $R_1$ contains that of $R_2$.
• Contracting edges in a way that preserves multicut feasibility, especially when all requests from a node share the same direction or are included in others.
• Leveraging the structure of caterpillars (where the internal tree is a path) to build intuition before generalizing to arbitrary trees.
• Ensuring each reduction rule is computable in polynomial time and that the number of rule applications is bounded by a polynomial in $k$.

Experimental results

Research questions

• RQ1 Does the Multicut in Trees problem admit a polynomial kernel, given that it is known to be FPT with an exponential kernel?
• RQ2 Can reduction rules based on request inclusion, disjointness, and directional consistency be used to safely shrink the instance to polynomial size in $k$?
• RQ3 Can the structural properties of caterpillars—such as wingspans and internal path dominance—be generalized to arbitrary trees to achieve kernelization?
• RQ4 Is there a set of polynomial-time computable reduction rules that preserve the solution and bound the kernel size to $O(k^6)$?

Key findings

• The paper establishes that Multicut in Trees admits a polynomial kernel of size $O(k^6)$, resolving an open problem in parameterized complexity.
• The kernel is constructed via a sequence of six safe reduction rules that preserve the solution and reduce the instance size to a polynomial in $k$.
• The reduction rules are applied in a fixed order and are computable in polynomial time, ensuring the kernelization process is efficient.
• The key insight is the use of wingspans and dominance relations to identify redundant leaves and edges that can be safely contracted.
• The result holds for the unweighted case; the authors note that their techniques do not extend to the weighted version of the problem.
• The proof is first developed for caterpillars and then generalized to arbitrary trees, demonstrating the robustness of the reduction rules.