Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree

"In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1,2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97^k n^4 log W) using O(n^3 log nW log k) space. This is the first single-exponential time, polynomial-space FPT algorithm for the weighted Steiner tree problem." PLEASE NOTE:This is an author-created version that the author has self-archived to the "Aaltodoc" (aaltodoc.aalto.fi) faculty-level repository at Aalto University. The final publication is available at link.springer.com via the link http://dx.doi.org/10.1007/978-3-662-47672-7_40Peer reviewe

Parameterized Complexity of Secluded Connectivity Problems

The Secluded Path problem models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. The problems were introduced by Chechik et al. in [ESA 2013]. Among other results, Chechik et al. have shown that Secluded Path is fixed-parameter tractable (FPT) on unweighted graphs being parameterized by the maximum vertex degree of the graph and that Secluded Steiner Tree is FPT parameterized by the treewidth of the graph. In this work, we obtain the following results about parameterized complexity of secluded connectivity problems. We give FPT-algorithms deciding if a graph G with a given cost function contains a secluded path and a secluded Steiner tree of exposure at most k with the cost at most C. We initiate the study of "above guarantee" parameterizations for secluded problems, where the lower bound is given by the size of a Steiner tree. We investigate Secluded Steiner Tree from kernelization perspective and provide several lower and upper bounds when parameters are the treewidth, the size of a vertex cover, maximum vertex degree and the solution size. Finally, we refine the algorithmic result of Chechik et al. by improving the exponential dependence from the treewidth of the input graph.Comment: Minor corrections are don

Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space

A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal

Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Dynamic programming on tree decompositions is a frequently used approach to solve otherwise intractable problems on instances of small treewidth. In recent work by Bodlaender et al., it was shown that for many connectivity problems, there exist algorithms that use time, linear in the number of vertices, and single exponential in the width of the tree decomposition that is used. The central idea is that it suffices to compute representative sets, and these can be computed efficiently with help of Gaussian elimination. In this paper, we give an experimental evaluation of this technique for the Steiner Tree problem. A comparison of the classic dynamic programming algorithm and the improved dynamic programming algorithm that employs the table reduction shows that the new approach gives significant improvements on the running time of the algorithm and the size of the tables computed by the dynamic programming algorithm, and thus that the rank based approach from Bodlaender et al. does not only give significant theoretical improvements but also is a viable approach in a practical setting, and showcases the potential of exploiting the idea of representative sets for speeding up dynamic programming algorithms

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted graph $G$ embedded on a surface of genus $g$ and a designated face $f$ bounded by a simple cycle of length $k$, uncovers a set $F \subseteq E(G)$ of size polynomial in $g$ and $k$ that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of $f$. We apply this general theorem to prove that: * given an unweighted graph $G$ embedded on a surface of genus $g$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal Steiner tree $T$ for $S$ and that has size polynomial in $g$ and $|E(T)|$; * an analogous result holds for an optimal Steiner forest for a set $S$ of terminal pairs; * given an unweighted planar graph $G$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal (edge) multiway cut $C$ separating $S$ and that has size polynomial in $|C|$. In the language of parameterized complexity, these results imply the first polynomial kernels for Steiner Tree and Steiner Forest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution