Core Concepts
Polynomial-time algorithms can detect 3-essential vertices for Vertex Multicut and 3.5-essential vertices for Cograph Deletion, enabling efficient fixed-parameter tractable algorithms for these problems.
Abstract
The paper presents several new results on the detection of essential vertices for two optimization problems on graphs: Vertex Multicut and Cograph Deletion.
For Vertex Multicut:
The authors provide a polynomial-time algorithm to detect 3-essential vertices. This means that all vertices in an optimal solution are either 3-essential or part of a small set of non-essential vertices.
Using this detection algorithm, they show that an optimal vertex multicut can be computed in time 2^O(ℓ^3) * n^O(1), where ℓ is the number of vertices in an optimal solution that are not 3-essential.
For the directed version of the problem, they provide a polynomial-time algorithm to detect 5-essential vertices.
For Cograph Deletion:
The authors give a polynomial-time algorithm to detect 3.5-essential vertices. This improves upon the best-known 4-approximation algorithm for the problem.
They also analyze the integrality gap of a standard linear programming relaxation for Cograph Deletion, proving it to be 4.
The positive results are obtained by analyzing the integrality gaps of certain linear programming relaxations associated with the problems. The authors also provide several hardness results, showing that for sufficiently small values of the essentiality threshold c, the detection task becomes NP-hard under the Unique Games Conjecture.