3 research outputs found
A Linear Time Parameterized Algorithm for Node Unique Label Cover
The optimization version of the Unique Label Cover problem is at the heart of
the Unique Games Conjecture which has played an important role in the proof of
several tight inapproximability results. In recent years, this problem has been
also studied extensively from the point of view of parameterized complexity.
Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable
(FPT) and Wahlstr\"om [SODA 2014] gave an FPT algorithm with an improved
parameter dependence. Subsequently, Iwata, Wahlstr\"om and Yoshida [2014]
proved that the edge version of Unique Label Cover can be solved in linear
FPT-time. That is, there is an FPT algorithm whose dependence on the input-size
is linear. However, such an algorithm for the node version of the problem was
left as an open problem. In this paper, we resolve this question by presenting
the first linear-time FPT algorithm for Node Unique Label Cover
Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms
We present two new combinatorial tools for the design of parameterized
algorithms. The first is a simple linear time randomized algorithm that given
as input a -degenerate graph and an integer , outputs an independent
set , such that for every independent set in of size at most ,
the probability that is a subset of is at least .The second is a new (deterministic) polynomial
time graph sparsification procedure that given a graph , a set of terminal pairs and an
integer , returns an induced subgraph of that maintains all
the inclusion minimal multicuts of of size at most , and does not
contain any -vertex connected set of size . In
particular, excludes a clique of size as a
topological minor. Put together, our new tools yield new randomized fixed
parameter tractable (FPT) algorithms for Stable - Separator, Stable Odd
Cycle Transversal and Stable Multicut on general graphs, and for Stable
Directed Feedback Vertex Set on -degenerate graphs, resolving two problems
left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our
algorithms can be derandomized at the cost of a small overhead in the running
time.Comment: 35 page