2,745 research outputs found
On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion
In this paper, we investigate the approximability of two node deletion
problems. Given a vertex weighted graph and a specified, or
"distinguished" vertex , MDD(min) is the problem of finding a minimum
weight vertex set such that becomes the
minimum degree vertex in ; and MDD(max) is the problem of
finding a minimum weight vertex set such that
becomes the maximum degree vertex in . These are known
-complete problems and have been studied from the parameterized complexity
point of view in previous work. Here, we prove that for any ,
both the problems cannot be approximated within a factor , unless . We also show that for any
, MDD(min) cannot be approximated within a factor on bipartite graphs, unless , and that for any , MDD(max) cannot be approximated within a
factor on bipartite graphs, unless . We give an factor approximation algorithm
for MDD(max) on general graphs, provided the degree of is . We
then show that if the degree of is , a similar result holds
for MDD(min). We prove that MDD(max) is -complete on 3-regular unweighted
graphs and provide an approximation algorithm with ratio when is a
3-regular unweighted graph. In addition, we show that MDD(min) can be solved in
polynomial time when is a regular graph of constant degree.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete
Algorithm
Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees
Let be an -node and -edge positively real-weighted undirected
graph. For any given integer , we study the problem of designing a
sparse \emph{f-edge-fault-tolerant} (-EFT) {\em -approximate
single-source shortest-path tree} (-ASPT), namely a subgraph of
having as few edges as possible and which, following the failure of a set
of at most edges in , contains paths from a fixed source that are
stretched at most by a factor of . To this respect, we provide an
algorithm that efficiently computes an -EFT -ASPT of size . Our structure improves on a previous related construction designed for
\emph{unweighted} graphs, having the same size but guaranteeing a larger
stretch factor of , plus an additive term of .
Then, we show how to convert our structure into an efficient -EFT
\emph{single-source distance oracle} (SSDO), that can be built in
time, has size , and is able to report,
after the failure of the edge set , in time a
-approximate distance from the source to any node, and a
corresponding approximate path in the same amount of time plus the path's size.
Such an oracle is obtained by handling another fundamental problem, namely that
of updating a \emph{minimum spanning forest} (MSF) of after that a
\emph{batch} of simultaneous edge modifications (i.e., edge insertions,
deletions and weight changes) is performed. For this problem, we build in time a \emph{sensitivity} oracle of size , that
reports in time the (at most ) edges either exiting from
or entering into the MSF. [...]Comment: 16 pages, 4 figure
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy
The principle of maximum entropy (Maxent) is often used to obtain prior
probability distributions as a method to obtain a Gibbs measure under some
restriction giving the probability that a system will be in a certain state
compared to the rest of the elements in the distribution. Because classical
entropy-based Maxent collapses cases confounding all distinct degrees of
randomness and pseudo-randomness, here we take into consideration the
generative mechanism of the systems considered in the ensemble to separate
objects that may comply with the principle under some restriction and whose
entropy is maximal but may be generated recursively from those that are
actually algorithmically random offering a refinement to classical Maxent. We
take advantage of a causal algorithmic calculus to derive a thermodynamic-like
result based on how difficult it is to reprogram a computer code. Using the
distinction between computable and algorithmic randomness we quantify the cost
in information loss associated with reprogramming. To illustrate this we apply
the algorithmic refinement to Maxent on graphs and introduce a Maximal
Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a
generalisation over previous approaches. We discuss practical implications of
evaluation of network randomness. Our analysis provides insight in that the
reprogrammability asymmetry appears to originate from a non-monotonic
relationship to algorithmic probability. Our analysis motivates further
analysis of the origin and consequences of the aforementioned asymmetries,
reprogrammability, and computation.Comment: 30 page
Explicit linear kernels via dynamic programming
Several algorithmic meta-theorems on kernelization have appeared in the last
years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of
bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding
a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed
topological minor. Typically, these results guarantee the existence of linear
or polynomial kernels on sparse graph classes for problems satisfying some
generic conditions but, mainly due to their generality, it is not clear how to
derive from them constructive kernels with explicit constants. In this paper we
make a step toward a fully constructive meta-kernelization theory on sparse
graphs. Our approach is based on a more explicit protrusion replacement
machinery that, instead of expressibility in CMSO logic, uses dynamic
programming, which allows us to find an explicit upper bound on the size of the
derived kernels. We demonstrate the usefulness of our techniques by providing
the first explicit linear kernels for -Dominating Set and -Scattered Set
on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs
excluding a fixed (topological) minor in the case where all the graphs in
\mathcal{F} are connected.Comment: 32 page
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