6 research outputs found
Lossy Kernels for Graph Contraction Problems
We study some well-known graph contraction problems in the recently introduced framework of lossy kernelization. In classical kernelization, given an instance (I,k) of a parameterized problem, we are interested in obtaining (in polynomial time) an equivalent instance (I\u27,k\u27) of the same problem whose size is bounded by a function in k. This notion however has a major limitation. Given an approximate solution to the instance (I\u27,k\u27), we can say nothing about the original instance (I,k). To handle this issue, among others, the framework of lossy kernelization was introduced. In this framework, for a constant alpha, given an instance (I,k) we obtain an instance (I\u27,k\u27) of the same problem such that, for every c>1, any c-approximate solution to (I\u27,k\u27) can be turned into a (c*alpha)-approximate solution to the original instance (I, k) in polynomial time. Naturally, we are interested in a polynomial time algorithm for this task, and further require that |I\u27| + k\u27 = k^{O(1)}. Akin to the notion of polynomial time approximation schemes in approximation algorithms, a parameterized problem is said to admit a polynomial size approximate kernelization scheme (PSAKS) if it admits a polynomial size alpha-approximate kernel for every approximation parameter alpha > 1. In this work, we design PSAKSs for Tree Contraction, Star Contraction, Out-Tree Contraction and Cactus Contraction problems. These problems do not admit polynomial kernels, and we show that each of them admit a PSAKS with running time k^{f(alpha)}|I|^{O(1)} that returns an instance of size k^{g(alpha)} where f(alpha) and g(alpha) are constants depending on alpha
Lossy Kernelization of Same-Size Clustering
In this work, we study the k-median clustering problem with an additional equal-size constraint on the clusters from the perspective of parameterized preprocessing. Our main result is the first lossy (2-approximate) polynomial kernel for this problem parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existence of an (exact) kernel of polynomial size and a PTAS
Parameterized Approximation Algorithms for Bidirected Steiner Network Problems
The Directed Steiner Network (DSN) problem takes as input a directed
edge-weighted graph and a set of
demand pairs. The aim is to compute the cheapest network for
which there is an path for each . It is known
that this problem is notoriously hard as there is no
-approximation algorithm under Gap-ETH, even when parametrizing
the runtime by [Dinur & Manurangsi, ITCS 2018]. In light of this, we
systematically study several special cases of DSN and determine their
parameterized approximability for the parameter .
For the bi-DSN problem, the aim is to compute a planar
optimum solution in a bidirected graph , i.e., for every edge
of the reverse edge exists and has the same weight. This problem
is a generalization of several well-studied special cases. Our main result is
that this problem admits a parameterized approximation scheme (PAS) for . We
also prove that our result is tight in the sense that (a) the runtime of our
PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists
for any generalization of bi-DSN, unless FPT=W[1].
One important special case of DSN is the Strongly Connected Steiner Subgraph
(SCSS) problem, for which the solution network needs to strongly
connect a given set of terminals. It has been observed before that for SCSS
a parameterized -approximation exists when parameterized by [Chitnis et
al., IPEC 2013]. We give a tight inapproximability result by showing that for
no parameterized -approximation algorithm exists under
Gap-ETH. Additionally we show that when restricting the input of SCSS to
bidirected graphs, the problem remains NP-hard but becomes FPT for