Hitting Subgraphs in Sparse Graphs and Geometric Intersection Graphs

We investigate a fundamental vertex-deletion problem called (Induced) Subgraph Hitting: given a graph $G$ and a set $\mathcal{F}$ of forbidden graphs, the goal is to compute a minimum-sized set $S$ of vertices of $G$ such that $G-S$ does not contain any graph in $\mathcal{F}$ as an (induced) subgraph. This is a generic problem that encompasses many well-known problems that were extensively studied on their own, particularly (but not only) from the perspectives of both approximation and parameterization. We focus on the design of efficient approximation schemes, i.e., with running time $f(\varepsilon,\mathcal{F}) \cdot n^{O(1)}$, which are also of significant interest to both communities. Technically, our main contribution is a linear-time approximation-preserving reduction from (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of bounded expansion to the same problem on bounded degree graphs within $\mathcal{G}$. This yields a novel algorithmic technique to design (efficient) approximation schemes for the problem on very broad graph classes, well beyond the state-of-the-art. Specifically, applying this reduction, we derive approximation schemes with (almost) linear running time for the problem on any graph classes that have strongly sublinear separators and many important classes of geometric intersection graphs (such as fat-object graphs, pseudo-disk graphs, etc.). Our proofs introduce novel concepts and combinatorial observations that may be of independent interest (and, which we believe, will find other uses) for studies of approximation algorithms, parameterized complexity, sparse graph classes, and geometric intersection graphs. As a byproduct, we also obtain the first robust algorithm for $k$-Subgraph Isomorphism on intersection graphs of fat objects and pseudo-disks, with running time $f(k) \cdot n \log n + O(m)$.Comment: 60 pages, abstract shortened to fulfill the length limi

The Parameterized Complexity of Degree Constrained Editing Problems

This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are within the given constraints. Parameterized complexity gives a framework for examining problems that are traditionally considered intractable and developing efficient exact algorithms for them, or showing that it is unlikely that they have such algorithms, by introducing an additional component to the input, the parameter, which gives additional information about the structure of the problem. If the problem has an algorithm that is exponential in the parameter, but polynomial, with constant degree, in the size of the input, then it is considered to be fixed-parameter tractable. Parameterized complexity also provides an intractability framework for identifying problems that are likely to not have such an algorithm. Degree constrained editing problems provide natural parameterizations in terms of the total cost k of vertex deletions, edge deletions and edge additions allowed, and the upper bound r on the degree of the vertices remaining after editing. We define a class of degree constrained editing problems, WDCE, which generalises several well know problems, such as Degree r Deletion, Cubic Subgraph, r-Regular Subgraph, f-Factor and General Factor. We show that in general if both k and r are part of the parameter, problems in the WDCE class are fixed-parameter tractable, and if parameterized by k or r alone, the problems are intractable in a parameterized sense. We further show cases of WDCE that have polynomial time kernelizations, and in particular when all the degree constraints are a single number and the editing operations include vertex deletion and edge deletion we show that there is a kernel with at most O(kr(k + r)) vertices. If we allow vertex deletion and edge addition, we show that despite remaining fixed-parameter tractable when parameterized by k and r together, the problems are unlikely to have polynomial sized kernelizations, or polynomial time kernelizations of a certain form, under certain complexity theoretic assumptions. We also examine a more general case where given an input graph the question is whether with at most k deletions the graph can be made r-degenerate. We show that in this case the problems are intractable, even when r is a constant

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum