1 research outputs found
On the Feasibility of Maintenance Algorithms in Dynamic Graphs
Near ubiquitous mobile computing has led to intense interest in dynamic graph
theory. This provides a new and challenging setting for algorithmics and
complexity theory. For any graph-based problem, the rapid evolution of a
(possibly disconnected) graph over time naturally leads to the important
complexity question: is it better to calculate a new solution from scratch or
to adapt the known solution on the prior graph to quickly provide a solution of
guaranteed quality for the changed graph?
In this paper, we demonstrate that the former is the best approach in some
cases, but that there are cases where the latter is feasible. We prove that,
under certain conditions, hard problems cannot even be approximated in any
reasonable complexity bound --- i.e., even with a large amount of time, having
a solution to a very similar graph does not help in computing a solution to the
current graph. To achieve this, we formalize the idea as a maintenance
algorithm. Using r-Regular Subgraph as the primary example we show that
W[1]-hardness for the parameterized approximation problem implies the
non-existence of a maintenance algorithm for the given approximation ratio.
Conversely we show that Vertex Cover, which is fixed-parameter tractable, has a
2-approximate maintenance algorithm. The implications of NP-hardness and
NPO-hardness are also explored