Single-Exponential FPT Algorithms for Enumerating Secluded F\mathcal{F}-Free Subgraphs and Deleting to Scattered Graph Classes

The celebrated notion of important separators bounds the number of small $(S,T)$-separators in a graph which are 'farthest from $S$' in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive that is phrased in terms of $k$-secluded vertex sets: sets with an open neighborhood of size at most $k$. In this terminology, the bound on important separators says that there are at most $4^k$ maximal $k$-secluded connected vertex sets $C$ containing $S$ but disjoint from $T$. We generalize this statement significantly: even when we demand that $G[C]$ avoids a finite set $\mathcal{F}$ of forbidden induced subgraphs, the number of such maximal subgraphs is $2^{O(k)}$ and they can be enumerated efficiently. This allows us to make significant improvements for two problems from the literature. Our first application concerns the 'Connected $k$-Secluded $\mathcal{F}$-free subgraph' problem, where $\mathcal{F}$ is a finite set of forbidden induced subgraphs. Given a graph in which each vertex has a positive integer weight, the problem asks to find a maximum-weight connected $k$-secluded vertex set $C \subseteq V(G)$ such that $G[C]$ does not contain an induced subgraph isomorphic to any $F \in \mathcal{F}$. The parameterization by $k$ is known to be solvable in triple-exponential time via the technique of recursive understanding, which we improve to single-exponential. Our second application concerns the deletion problem to scattered graph classes. Here, the task is to find a vertex set of size at most $k$ whose removal yields a graph whose each connected component belongs to one of the prescribed graph classes $\Pi_1, \ldots, \Pi_d$. We obtain a single-exponential algorithm whenever each class $\Pi_i$ is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.Comment: To appear at ISAAC'2

5-Approximation for H\mathcal{H}-Treewidth Essentially as Fast as H\mathcal{H}-Deletion Parameterized by Solution Size

The notion of $\mathcal{H}$-treewidth, where $\mathcal{H}$ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of $\mathcal{H}$-treewidth at most $k$ can be decomposed into (arbitrarily large) $\mathcal{H}$-subgraphs which interact only through vertex sets of size $O(k)$ which can be organized in a tree-like fashion. $\mathcal{H}$-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for $\mathcal{H}$-deletion problems, which ask to find a minimum vertex set whose removal from a given graph $G$ turns it into a member of $\mathcal{H}$. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree $\mathcal{H}$-decompositions. We present FPT approximation algorithms to compute tree $\mathcal{H}$-decompositions for hereditary and union-closed graph classes $\mathcal{H}$. Given a graph of $\mathcal{H}$-treewidth $k$, we can compute a 5-approximate tree $\mathcal{H}$-decomposition in time $f(O(k)) \cdot n^{O(1)}$ whenever $\mathcal{H}$-deletion parameterized by solution size can be solved in time $f(k) \cdot n^{O(1)}$ for some function $f(k) \geq 2^k$. The current-best algorithms either achieve an approximation factor of $k^{O(1)}$ or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time $2^{O(k)} \cdot n^{O(1)}$ parameterized by $\mathsf{bipartite}$-treewidth and Vertex Planarization in time $2^{O(k \log k)} \cdot n^{O(1)}$ parameterized by $\mathsf{planar}$-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.Comment: Conference version to appear at the European Symposium on Algorithms (ESA 2023

5-Approximation for ?-Treewidth Essentially as Fast as ?-Deletion Parameterized by Solution Size

The notion of ?-treewidth, where ? is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of ?-treewidth at most k can be decomposed into (arbitrarily large) ?-subgraphs which interact only through vertex sets of size ?(k) which can be organized in a tree-like fashion. ?-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for ?-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of ?. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree ?-decompositions. We present FPT-approximation algorithms to compute tree ?-decompositions for hereditary and union-closed graph classes ?. Given a graph of ?-treewidth k, we can compute a 5-approximate tree ?-decomposition in time f(?(k)) ? n^?(1) whenever ?-deletion parameterized by solution size can be solved in time f(k) ? n^?(1) for some function f(k) ? 2^k. The current-best algorithms either achieve an approximation factor of k^?(1) or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^?(k) ? n^?(1) parameterized by bipartite-treewidth and Vertex Planarization in time 2^?(k log k) ? n^?(1) parameterized by planar-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures

Search-Space Reduction via Essential Vertices

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution

Preprocessing Vertex-Deletion Problems: Characterizing Graph Properties by Low-Rank Adjacencies

We consider the ?-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties ?. Given an input graph G, this problem asks whether there is a subset of at most k vertices whose removal ensures the resulting graph does not contain a graph from ? as induced subgraph. Many vertex-deletion problems such as Perfect Deletion, Wheel-free Deletion, and Interval Deletion fit into this framework. We introduce the concept of characterizing a graph property ? by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for ?-Free Deletion parameterized by the size of a vertex cover. The resulting framework captures problems such as AT-Free Deletion, Wheel-free Deletion, and Interval Deletion. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. [JCSS 2014]. Our main technical contribution shows how linear-algebraic dependence of suitably defined vectors over ?? implies graph-theoretic statements about the presence of forbidden induced subgraphs