891 research outputs found
(Non-)existence of Polynomial Kernels for the Test Cover Problem
The input of the Test Cover problem consists of a set of vertices, and a
collection of distinct subsets of , called
tests. A test separates a pair of vertices if A subcollection is a test cover if each
pair of distinct vertices is separated by a test in . The
objective is to find a test cover of minimum cardinality, if one exists. This
problem is NP-hard.
We consider two parameterizations the Test Cover problem with parameter :
(a) decide whether there is a test cover with at most tests, (b) decide
whether there is a test cover with at most tests. Both
parameterizations are known to be fixed-parameter tractable. We prove that none
have a polynomial size kernel unless . Our proofs use
the cross-composition method recently introduced by Bodlaender et al. (2011)
and parametric duality introduced by Chen et al. (2005). The result for the
parameterization (a) was an open problem (private communications with Henning
Fernau and Jiong Guo, Jan.-Feb. 2012). We also show that the parameterization
(a) admits a polynomial size kernel if the size of each test is upper-bounded
by a constant
Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)HHHGc_i(u)u\in V(G)i\in V(H)GH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
- β¦