132,264 research outputs found
Solution discovery via reconfiguration for problems in P
In the recently introduced framework of solution discovery via
reconfiguration [Fellows et al., ECAI 2023], we are given an initial
configuration of tokens on a graph and the question is whether we can
transform this configuration into a feasible solution (for some problem) via a
bounded number of small modification steps. In this work, we study solution
discovery variants of polynomial-time solvable problems, namely Spanning Tree
Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut
Discovery in the unrestricted token addition/removal model, the token jumping
model, and the token sliding model. In the unrestricted token addition/removal
model, we show that all four discovery variants remain in P. For the toking
jumping model we also prove containment in P, except for Vertex/Edge Cut
Discovery, for which we prove NP-completeness. Finally, in the token sliding
model, almost all considered problems become NP-complete, the exception being
Spanning Tree Discovery, which remains polynomial-time solvable. We then study
the parameterized complexity of the NP-complete problems and provide a full
classification of tractability with respect to the parameters solution size
(number of tokens) and transformation budget (number of steps) . Along
the way, we observe strong connections between the solution discovery variants
of our base problems and their (weighted) rainbow variants as well as their
red-blue variants with cardinality constraints
On the (non-)existence of polynomial kernels for Pl-free edge modification problems
Given a graph G = (V,E) and an integer k, an edge modification problem for a
graph property P consists in deciding whether there exists a set of edges F of
size at most k such that the graph H = (V,E \vartriangle F) satisfies the
property P. In the P edge-completion problem, the set F of edges is constrained
to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no
constraint is imposed on F in the P edge-edition problem. A number of
optimization problems can be expressed in terms of graph modification problems
which have been extensively studied in the context of parameterized complexity.
When parameterized by the size k of the edge set F, it has been proved that if
P is an hereditary property characterized by a finite set of forbidden induced
subgraphs, then the three P edge-modification problems are FPT. It was then
natural to ask whether these problems also admit a polynomial size kernel.
Using recent lower bound techniques, Kratsch and Wahlstrom answered this
question negatively. However, the problem remains open on many natural graph
classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom
asked whether the result holds when the forbidden subgraphs are paths or cycles
and pointed out that the problem is already open in the case of P4-free graphs
(i.e. cographs). This paper provides positive and negative results in that line
of research. We prove that parameterized cograph edge modification problems
have cubic vertex kernels whereas polynomial kernels are unlikely to exist for
the Pl-free and Cl-free edge-deletion problems for large enough l
On the classification of automorphisms of trees
We identify the complexity of the classification problem for automorphisms of
a given countable regularly branching tree up to conjugacy. We consider both
the rooted and unrooted cases. Additionally, we calculate the complexity of the
conjugacy problem in the case of automorphisms of several non-regularly
branching trees
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