Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter

Wireless Communication Technologies for Safe Cooperative Cyber Physical Systems

Cooperative Cyber-Physical Systems (Co-CPSs) can be enabled using wireless communication technologies, which in principle should address reliability and safety challenges. Safety for Co-CPS enabled by wireless communication technologies is a crucial aspect and requires new dedicated design approaches. In this paper, we provide an overview of five Co-CPS use cases, as introduced in our SafeCOP EU project, and analyze their safety design requirements. Next, we provide a comprehensive analysis of the main existing wireless communication technologies giving details about the protocols developed within particular standardization bodies. We also investigate to what extent they address the non-functional requirements in terms of safety, security and real time, in the different application domains of each use case. Finally, we discuss general recommendations about the use of different wireless communication technologies showing their potentials in the selected real-world use cases. The discussion is provided under consideration in the 5G standardization process within 3GPP, whose current efforts are inline to current gaps in wireless communications protocols for Co-CPSs including many future use casesinfo:eu-repo/semantics/publishedVersio

An Empirical Evaluation of k-Means Coresets

Coresets are among the most popular paradigms for summarizing data. In particular, there exist many high performance coresets for clustering problems such as k-means in both theory and practice. Curiously, there exists no work on comparing the quality of available k-means coresets. In this paper we perform such an evaluation. There currently is no algorithm known to measure the distortion of a candidate coreset. We provide some evidence as to why this might be computationally difficult. To complement this, we propose a benchmark for which we argue that computing coresets is challenging and which also allows us an easy (heuristic) evaluation of coresets. Using this benchmark and real-world data sets, we conduct an exhaustive evaluation of the most commonly used coreset algorithms from theory and practice

Treedepth Parameterized by Vertex Cover Number

To solve hard graph problems from the parameterized perspective, structural parameters have commonly been used. In particular, vertex cover number is frequently used in this context. In this paper, we study the problem of computing the treedepth of a given graph G. We show that there are an O(tau(G)^3) vertex kernel and an O(4^{tau(G)}*tau(G)*n) time fixed-parameter algorithm for this problem, where tau(G) is the size of a minimum vertex cover of G and n is the number of vertices of G

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

Minimal Absent Words in Rooted and Unrooted Trees

We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet of cardinality. We show that the set of minimal absent words of a rooted (resp. unrooted) tree T with n nodes has cardinality (resp.), and we show that these bounds are realized. Then, we exhibit algorithms to compute all minimal absent words in a rooted (resp. unrooted) tree in output-sensitive time (resp. assuming an integer alphabet of size polynomial in n

SimBa: An Efficient Tool for Approximating Rips-Filtration Persistence via Simplicial Batch-Collapse

In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on P indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short of scaling well, especially for high dimensional data. In this paper, we investigate the advantages and limitations of existing approaches. Based on insights gained from the experiments, we propose an efficient new algorithm, called SimBa, for approximating the persistent homology of Rips filtrations with quality guarantees. Our new algorithm leverages a batch collapse strategy as well as a new sparse Rips-like filtration. We experiment on a variety of low and high dimensional data sets. We show that our strategy presents a significant size reduction, and our algorithm for approximating Rips filtration persistence is order of magnitude faster than existing methods in practice