14 research outputs found
Metric Dimension for Gabriel Unit Disk Graphs is NP-Complete
We show that finding a minimal number of landmark nodes for a unique virtual
addressing by hop-distances in wireless ad-hoc sensor networks is NP-complete
even if the networks are unit disk graphs that contain only Gabriel edges. This
problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The
Gabriel edges of a unit disc graph induce a planar O(\sqrt{n}) distance and an
optimal energy spanner. This is one of the most interesting restrictions of
Metric Dimension in the context of wireless multi-hop networks.Comment: A brief announcement of this result has been published in the
proceedings of ALGOSENSORS 201
Metric Dimension of Amalgamation of Graphs
A set of vertices resolves a graph if every vertex is uniquely
determined by its vector of distances to the vertices in . The metric
dimension of is the minimum cardinality of a resolving set of .
Let be a finite collection of graphs and each
has a fixed vertex or a fixed edge called a terminal
vertex or edge, respectively. The \emph{vertex-amalgamation} of , denoted by , is formed by taking all
the 's and identifying their terminal vertices. Similarly, the
\emph{edge-amalgamation} of , denoted by
, is formed by taking all the 's and identifying
their terminal edges.
Here we study the metric dimensions of vertex-amalgamation and
edge-amalgamation for finite collection of arbitrary graphs. We give lower and
upper bounds for the dimensions, show that the bounds are tight, and construct
infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on
Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised
version 21 December 201
On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs
Numerous problems consisting in identifying vertices in graphs using
distances are useful in domains such as network verification and graph
isomorphism. Unifying them into a meta-problem may be of main interest. We
introduce here a promising solution named Distance Identifying Set. The model
contains Identifying Code (IC), Locating Dominating Set (LD) and their
generalizations -IC and -LD where the closed neighborhood is considered
up to distance . It also contains Metric Dimension (MD) and its refinement
-MD in which the distance between two vertices is considered as infinite if
the real distance exceeds . Note that while IC = 1-IC and LD = 1-LD, we have
MD = -MD; we say that MD is not local
In this article, we prove computational lower bounds for several problems
included in Distance Identifying Set by providing generic reductions from
(Planar) Hitting Set to the meta-problem. We mainly focus on two families of
problem from the meta-problem: the first one, called bipartite gifted local,
contains -IC, -LD and -MD for each positive integer while the
second one, called 1-layered, contains LD, MD and -MD for each positive
integer . We have:
- the 1-layered problems are NP-hard even in bipartite apex graphs,
- the bipartite gifted local problems are NP-hard even in bipartite planar
graphs,
- assuming ETH, all these problems cannot be solved in when
restricted to bipartite planar or apex graph, respectively, and they cannot be
solved in on bipartite graphs,
- even restricted to bipartite graphs, they do not admit parameterized
algorithms in except if W[0] = W[2]. Here is the
solution size of a relevant identifying set.
In particular, Metric Dimension cannot be solved in under ETH,
answering a question of Hartung in 2013
Metric Dimension Parameterized by Treewidth
A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polytime algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth.
We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time f(pw)n^{o(pw)} on n-vertex graphs of constant degree, with pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. [SIAM J. Discrete Math. \u2717] with respect to the combined parameter tl+Delta, where tl is the tree-length and Delta the maximum-degree of the input graph
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
Metric Dimension Parameterized By Treewidth
A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The METRIC DIMENSION problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. METRIC DIMENSION has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of METRIC DIMENSION with respect to treewidth. We provide a first answer to the question. We show that METRIC DIMENSION parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving METRIC DIMENSION in time f(pw)no(pw) on n-vertex graphs of constant degree, with pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter tl+Δ, where tl is the tree-length and Δ the maximum-degree of the input graph.publishedVersio