6,328 research outputs found
Parameterized and approximation complexity of the detection pair problem in graphs
We study the complexity of the problem DETECTION PAIR. A detection pair of a
graph is a pair of sets of detectors with , the
watchers, and , the listeners, such that for every pair
of vertices that are not dominated by a watcher of , there is a listener of
whose distances to and to are different. The goal is to minimize
. This problem generalizes the two classic problems DOMINATING SET and
METRIC DIMENSION, that correspond to the restrictions and
, respectively. DETECTION PAIR was recently introduced by Finbow,
Hartnell and Young [A. S. Finbow, B. L. Hartnell and J. R. Young. The
complexity of monitoring a network with both watchers and listeners.
Manuscript, 2015], who proved it to be NP-complete on trees, a surprising
result given that both DOMINATING SET and METRIC DIMENSION are known to be
linear-time solvable on trees. It follows from an existing reduction by Hartung
and Nichterlein for METRIC DIMENSION that even on bipartite subcubic graphs of
arbitrarily large girth, DETECTION PAIR is NP-hard to approximate within a
sub-logarithmic factor and W[2]-hard (when parameterized by solution size). We
show, using a reduction to SET COVER, that DETECTION PAIR is approximable
within a factor logarithmic in the number of vertices of the input graph. Our
two main results are a linear-time -approximation algorithm and an FPT
algorithm for DETECTION PAIR on trees.Comment: 13 page
Approximating Subdense Instances of Covering Problems
We study approximability of subdense instances of various covering problems
on graphs, defined as instances in which the minimum or average degree is
Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We
design new approximation algorithms as well as new polynomial time
approximation schemes (PTASs) for those problems and establish first
approximation hardness results for them. Interestingly, in some cases we were
able to prove optimality of the underlying approximation ratios, under usual
complexity-theoretic assumptions. Our results for the Vertex Cover problem
depend on an improved recursive sampling method which could be of independent
interest
Improved approximation for 3-dimensional matching via bounded pathwidth local search
One of the most natural optimization problems is the k-Set Packing problem,
where given a family of sets of size at most k one should select a maximum size
subfamily of pairwise disjoint sets. A special case of 3-Set Packing is the
well known 3-Dimensional Matching problem. Both problems belong to the Karp`s
list of 21 NP-complete problems. The best known polynomial time approximation
ratio for k-Set Packing is (k + eps)/2 and goes back to the work of Hurkens and
Schrijver [SIDMA`89], which gives (1.5 + eps)-approximation for 3-Dimensional
Matching. Those results are obtained by a simple local search algorithm, that
uses constant size swaps.
The main result of the paper is a new approach to local search for k-Set
Packing where only a special type of swaps is considered, which we call swaps
of bounded pathwidth. We show that for a fixed value of k one can search the
space of r-size swaps of constant pathwidth in c^r poly(|F|) time. Moreover we
present an analysis proving that a local search maximum with respect to O(log
|F|)-size swaps of constant pathwidth yields a polynomial time (k + 1 +
eps)/3-approximation algorithm, improving the best known approximation ratio
for k-Set Packing. In particular we improve the approximation ratio for
3-Dimensional Matching from 3/2 + eps to 4/3 + eps.Comment: To appear in proceedings of FOCS 201
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