Fixed-parameter tractability of multicut parameterized by the size of the cutset (1010.3633v3)

Abstract: Given an undirected graph $G$, a collection ${(s_1,t_1),..., (s_k,t_k)}$ of pairs of vertices, and an integer $p$, the Edge Multicut problem ask if there is a set $S$ of at most $p$ edges such that the removal of $S$ disconnects every $s_i$ from the corresponding $t_i$. Vertex Multicut is the analogous problem where $S$ is a set of at most $p$ vertices. Our main result is that both problems can be solved in time $2^{O(p^3)}... n^{O(1)}$, i.e., fixed-parameter tractable parameterized by the size $p$ of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form $f(p)... n^{O(1)}$ exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset.

Fixed-Parameter Tractability of Multicut Parameterized by the Size of the Cutset

This paper investigates the Edge Multicut and Vertex Multicut problems within the framework of parameterized complexity. The primary contribution is establishing the fixed-parameter tractability (FPT) of these problems when parameterized by the size of the cutset. The paper provides a formal proof that both Edge Multicut and Vertex Multicut problems are solvable in time $2^{O(p^3)}\cdot n^{O(1)}$, where $p$ is the parameter representing the size of the cutset. This result closes an open question in the parameterized complexity domain, demonstrating that solutions with small cutsets can be found efficiently.

Problem Context and Importance

The paper of cut problems in graph theory has a rich historical lineage, with foundational work such as Ford and Fulkerson's classical results on $s-t$ cuts and modern approximation algorithms for the sparsest cut problem. The Edge Multicut and Vertex Multicut problems extend this area, requiring the removal of a minimal set of edges or vertices to disconnect specified pairs of vertices. Although polynomial-time solvable for small instances, they become NP-hard as the size of the graphs increases, particularly when the number of vertex pairs ($k$) to be separated increases.

Main Results

A key achievement outlined is the FPT algorithm for the Edge and Vertex Multicut problems, parameterized by the size $p$ of the solution cutset. The authors articulate that the running time grows exponentially with $p^3$, but remains polynomial concerning the size of the graph. This implies that for small $p$, even very large graphs can be processed efficiently.

The authors contrast these results with the directed version of the Multicut problem, which they prove to be W[1]-hard. Thus, the fixed-parameter approach applicable to undirected graphs does not extend to directed graphs, reflecting a boundary in the tractability facilitated by problem structure.

Techniques and Algorithmic Framework

The paper introduces several innovative techniques for tackling the Multicut problem. The approach involves iterative compression and randomized sampling, culminating in efficient enumerations of important separators—a concept foundational to the parameterized complexity analysis. Central to this method is the iterative reduction process, which compresses solutions and operates over shadowless solutions—a novel reduction style that enhances the tractability of the problem.

The random sampling approach for important separators guarantees capturing close-to-optimal solutions by reducing the graph iteratively, allowing the problem to be tackled via known techniques such as Almost 2SAT.

Implications and Future Directions

The implications are significant in theoretical computer science and practical applications. Understanding that such complex separation problems are tractable under realistic assumptions affirms the strengths of parameterized complexity as a tool for real-world problem-solving. However, the complexity difference in directed graphs highlights open challenges, leading to new pathways for exploring similar problems in directed graphs and extending these techniques.

Future avenues include refining these algorithms to lower the dependency on $p^3$ in the time complexity and exploring heuristic methods inspired by the fixed-parameter properties identified. Exploration in dynamic graphs and real-time applications could materialize from this foundational work, advancing both theoretical insights and practical capabilities.

In summary, this research paper offers substantial contributions to the understanding of Multicut problems and establishes foundational techniques for ongoing paper in parameterized complexity. It presents a robust method to tackle graph separation tasks, with its implications traversing theoretical bounds and practical feasibility.