10 research outputs found
Hitting Subgraphs in Sparse Graphs and Geometric Intersection Graphs
We investigate a fundamental vertex-deletion problem called (Induced)
Subgraph Hitting: given a graph and a set of forbidden
graphs, the goal is to compute a minimum-sized set of vertices of such
that does not contain any graph in 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
, 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 of bounded expansion to the same problem on
bounded degree graphs within . 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 -Subgraph Isomorphism on intersection graphs of fat objects and
pseudo-disks, with running time .Comment: 60 pages, abstract shortened to fulfill the length limi
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Compound Logics for Modification Problems
We introduce a novel model-theoretic framework inspired from graph
modification and based on the interplay between model theory and algorithmic
graph minors. The core of our framework is a new compound logic operating with
two types of sentences, expressing graph modification: the modulator sentence,
defining some property of the modified part of the graph, and the target
sentence, defining some property of the resulting graph. In our framework,
modulator sentences are in counting monadic second-order logic (CMSOL) and have
models of bounded treewidth, while target sentences express first-order logic
(FOL) properties along with minor-exclusion. Our logic captures problems that
are not definable in first-order logic and, moreover, may have instances of
unbounded treewidth. Also, it permits the modeling of wide families of problems
involving vertex/edge removals, alternative modulator measures (such as
elimination distance or -treewidth), multistage modifications, and
various cut problems. Our main result is that, for this compound logic,
model-checking can be done in quadratic time. All derived algorithms are
constructive and this, as a byproduct, extends the constructibility horizon of
the algorithmic applications of the Graph Minors theorem of Robertson and
Seymour. The proposed logic can be seen as a general framework to capitalize on
the potential of the irrelevant vertex technique. It gives a way to deal with
problem instances of unbounded treewidth, for which Courcelle's theorem does
not apply. The proof of our meta-theorem combines novel combinatorial results
related to the Flat Wall theorem along with elements of the proof of
Courcelle's theorem and Gaifman's theorem. We finally prove extensions where
the target property is expressible in FOL+DP, i.e., the enhancement of FOL with
disjoint-paths predicates