A single-exponential fixed-parameter algorithm for distance-hereditary vertex deletion

Abstract

Vertex deletion problems ask whether it is possible to delete at most k vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter tractable algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our result with matching asymptotic lower bounds based on the exponential time hypothesis. As an application of our algorithm, we show that a vertex deletion set to distance-hereditary graphs can be used as a parameter which allows single-exponential fixed-parameter tractable algorithms for classical NP-hard problems.

A polynomial kernel for $3$-leaf power deletion

“…Distance-hereditary Deletion is a problem of deciding whether a graph has a vertex set of size at most k whose deletion makes it distance-hereditary. Eiben, Ganian, and Kwon [17] presented a singleexponential fixed-parameter tractable algorithm for Distance-hereditary Deletion, that is an algorithm with running time Opc k ¨nOp1q q for input size n and some constant c. To obtain our first result, we observe that if an input graph G does not contain a small obstruction, that is a minimal induced subgraph of size at most 5 that is not a 3-leaf power, then G is a 3-leaf power if and only if G is distance-hereditary. Hence, after branching on small obstructions, we can use the algorithm by Eiben, Ganian, and Kwon [17] as a black-box.…”

“…We improve this further by showing that a single-exponential fixed-parameter tractable algorithm exists for 3-leaf Power Deletion. To do so, we use the following theorem of Eiben, Ganian, and Kwon [17].…”

Deleting, Eliminating and Decomposing to Hereditary Classes Are All FPT-Equivalent

“…Moreover, for various well-studied families of H, we immediately derive FPT algorithms for all combinations of Vertex Deletion to H, Elimination Distance to H, Treewidth Decomposition to H parameterized by any of mod H (G) ed H (G) and tw H (G), which are covered in Theorem 1.1. For instance, we can invoke this theorem using well-known FPT algorithms for Vertex Deletion to H for several families of graphs that are CMSO definable and closed under disjoint union, such as families defined by a finite number of forbidden connected (a) minors, or (b) topological minors, or (c) induced subgraphs, or (d) H being bipartite, chordal, proper-interval, interval, and distance-hereditary; to name a few [73,14,15,13,28,33,31,49,58,63,70,71,72,54]. Thus, Theorem 1.1 provides a unified understanding of many recent results and resolves the parameterized complexity of several questions left open in the literature.…”

“…Further, it is known that vertexminors can be expressed in CMSO, this together with the fact that we can test whether a graph H is a vertex-minor of G or not in f (|H|)n O (1) time on graphs of bounded rankwidth leads to the desired algorithm [22,Theorem 6.11]. It is also important to mention that for Vertex Deletion to H 1 , also known as the Distance-Hereditary Vertex-Deletion problem, there is a dedicated algorithm running in time 2 O(k) n O(1) [28]. For us, two properties of H η are important: (a) expressibility in CMSO and (b) being closed under disjoint union.…”

Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth

“…Basic examples of parameters based on modulators include the vertex cover number (a modulator to edgeless graphs) [21,26] and the feedback vertex set number (a modulator to forests) [4,38]. For dense graphs, modulators to graphs of rank-width 1 have been studied [17,43], and it is known that for every constant c one can find a modulator of size at most k to graphs of rank-width c (if such a modulator exists) in time f (k) • n [42]. However, the algorithmic applications of such modulators have remained largely unexplored up to this point.…”