6 research outputs found

    Lossy Kernels for Graph Contraction Problems

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    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

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    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

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    The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E)G=(V,E) and a set D⊆V×V\mathcal{D}\subseteq V\times V of kk demand pairs. The aim is to compute the cheapest network N⊆GN\subseteq G for which there is an s→ts\to t path for each (s,t)∈D(s,t)\in\mathcal{D}. It is known that this problem is notoriously hard as there is no k1/4−o(1)k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parametrizing the runtime by kk [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter kk. For the bi-DSNPlanar_\text{Planar} problem, the aim is to compute a planar optimum solution N⊆GN\subseteq G in a bidirected graph GG, i.e., for every edge uvuv of GG the reverse edge vuvu 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 kk. 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-DSNPlanar_\text{Planar}, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N⊆GN\subseteq G needs to strongly connect a given set of kk terminals. It has been observed before that for SCSS a parameterized 22-approximation exists when parameterized by kk [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for kk no parameterized (2−ε)(2-\varepsilon)-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 kk

    35th Symposium on Theoretical Aspects of Computer Science: STACS 2018, February 28-March 3, 2018, Caen, France

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