31 research outputs found
Alternative parameterizations of Metric Dimension
A set of vertices in a graph is called resolving if for any two
distinct , there is such that , where denotes the length of a shortest path
between and in the graph . The metric dimension of
is the minimum cardinality of a resolving set. The Metric Dimension problem,
i.e. deciding whether , is NP-complete even for interval
graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs)
from the lens of parameterized complexity. The problem parameterized by was
proved to be -hard by Hartung and Nichterlein (2013) and we study the
dual parameterization, i.e., the problem of whether
where is the order of . We prove that the dual parameterization admits
(a) a kernel with at most vertices and (b) an algorithm of runtime
Hartung and Nichterlein (2013) also observed that Metric
Dimension is fixed-parameter tractable when parameterized by the vertex cover
number of the input graph. We complement this observation by showing
that it does not admit a polynomial kernel even when parameterized by . Our reduction also gives evidence for non-existence of polynomial Turing
kernels
Path-Contractions, Edge Deletions and Connectivity Preservation
We study several problems related to graph modification problems under
connectivity constraints from the perspective of parameterized complexity: {\sc
(Weighted) Biconnectivity Deletion}, where we are tasked with deleting~
edges while preserving biconnectivity in an undirected graph, {\sc
Vertex-deletion Preserving Strong Connectivity}, where we want to maintain
strong connectivity of a digraph while deleting exactly~ vertices, and {\sc
Path-contraction Preserving Strong Connectivity}, in which the operation of
path contraction on arcs is used instead. The parameterized tractability of
this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open
question and we answer it here in the negative: both variants of preserving
strong connectivity are -hard. Preserving biconnectivity, on the
other hand, turns out to be fixed parameter tractable and we provide a
-algorithm that solves {\sc Weighted Biconnectivity
Deletion}. Further, we show that the unweighted case even admits a randomized
polynomial kernel. All our results provide further interesting data points for
the systematic study of connectivity-preservation constraints in the
parameterized setting
On the complexity landscape of connected f-factor problems
Let G be an undirected simple graph having n vertices and let f:V(G)→{0,…,n−1} be a function. An f-factor of G is a spanning subgraph H such that dH(v)=f(v) for every vertex v∈V(G). The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte (Can J Math 6(1954):347–352, 1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connectedf-factor is easily seen to generalize Hamiltonian Cycle and hence is NP-complete. In fact, the Connected f-Factor problem remains NP-complete even when we restrict f(v) to be at least nϵ for each vertex v and constant 0≤ϵ1, the problem is NP-intermediate
Backdoors for Linear Temporal Logic
In the present paper, we introduce the backdoor set approach into the field of temporal logic for the global fragment of linear temporal logic. We study the parameterized complexity of the satisfiability problem parameterized by the size of the backdoor. We distinguish between backdoor detection and evaluation of backdoors into the fragments of Horn and Krom formulas. Here we classify the operator fragments of globally-operators for past/future/always, and the combination of them. Detection is shown to be fixed-parameter tractable whereas the complexity of evaluation behaves differently. We show that for Krom formulas the problem is paraNP-complete. For Horn formulas, the complexity is shown to be either fixed parameter tractable or paraNP-complete depending on the considered operator fragment
Subspaces of some nuclear sequence spaces
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23700/1/0000671.pd