Matthias Bentert

Matthias Bentert, Chen Avin, Stefan Schmid

The performance of distributed applications often critically depends on the interconnecting network or more specifically on its throughput: how fast data can be carried across a network. Over the last years, great progress has been made in understanding demand-oblivious throughput: how fast a given demand matrix describing pairwise communication requirements can be served on a given network. However, surprisingly little is known today about the achievable demand-aware throughput: the throughput on a network topology which can be optimized toward the demand. Such demand-aware networks have recently gained popularity in datacenters and are enabled by emerging reconfigurable optical technologies. In this paper, we are interested in both the achievable demand-aware throughput bounds as well as in the computational complexity of finding a throughput-optimizing network topology. We take a systematic approach and investigate four variants of demand-aware throughput: we analyze, and derive bounds for, two definitions of throughput, the classic throughput usually considered in the literature, and a new generalized definition which we call weak throughput; for each of them, we consider two routing models, a direct one, where demand can only be served on a single hop, and a general one, where multi-hop routing is allowed. Our main result is a separation result which solves an open problem in the literature about the classic throughput definition, showing that demand-aware topologies can outperform demand-oblivious topologies even in the worst case: the demand-aware throughput asymptotically approaches at least 5/8, while it is known that the demand-oblivious throughput is n/(2n-1), which is roughly 1/2. In terms of computational complexity, we show that computing the demand-aware weak throughput is NP-hard, but computing the demand-aware (weak) direct throughput is polynomial-time solvable.

Matthias Bentert, Fedor v. Fomin, Petr A. Golovach et al.

We study extensions of the classic \emph{Line Cover} problem, which asks whether a set of $n$ points in the plane can be covered using $k$ lines. Line Cover is known to be NP-hard, and we focus on two natural generalizations. The first is \textbf{Line Clustering}, where the goal is to find $k$ lines minimizing the sum of squared distances from the input points to their nearest line. The second is \textbf{Hyperplane Cover}, which asks whether $n$ points in $\mathbb{R}^d$ can be covered by $k$ hyperplanes. We also study the more general \textbf{Projective Clustering} problem, which unifies both settings and has applications in machine learning, data analysis, and computational geometry. In this problem, one seeks $k$ affine subspaces of dimension $r$ that minimize the sum of squared distances from the given points in $\mathbb{R}^d$ to the nearest subspace. Our results reveal notable differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable when parameterized by $k$, we show that Line Clustering is W[1]-hard with respect to $k$ and does not admit an algorithm with running time $n^{o(k)}$ unless the Exponential Time Hypothesis fails. Hyperplane Cover has been known to be NP-hard since the 1980s, following work of Megiddo and Tamir, even for $d=2$, we show that it remains NP-hard even when $k=2$. Finally, we present an algorithm for Projective Clustering running in $n^{O(dk(r+1))}$ time. This bound matches our lower bound for Line Clustering and generalizes the classic algorithm for $k$-Means Clustering ($r=0$) by Inaba, Katoh, and Imai [SoCG 1994].

Matthias Bentert, Shay Kutten, Darya Melnyk et al.

The theoretical model behind the pigeon post as a link layer in a communication network was introduced by Shannon (under the guise of studying One-Time Pads for cryptography). That is, to send a one-hop message to $v$, a node $u$ needs a mail pigeon bred and raised at $v$. When sending a message using a pigeon to $v$, node $u$ loses the pigeon. To send another message to $v$, node $u$ needs another pigeon of $v$. It has been demonstrated that the communication bandwidth achievable with pigeon post can exceed that of networks using other media. This has already motivated the introduction of Internet standards that allow the use of pigeons as Internet link-layer media. In this paper, we begin to fill in the missing piece: designing algorithms for breeding and scheduling pigeons to meet a given communication demand efficiently, minimizing the number of pigeons required. We consider singlehop, 2-hop, and multihop pigeon use. While the singlehop variant admits a simple characterization, both the 2-hop and the multihop variants are NP-hard. For the latter variants, we present a polynomial-time algorithm based on demand aggregation that achieves a 2-approximation for the number of pigeons used. We believe that this pigeon-based perspective offers both amusing and instructive insights into network design and hopefully, into ornithology.

Matthias Bentert, Fedor V. Fomin, Petr A. Golovach

Many problems are known to be solvable in subexponential parameterized time when the input graph is planar. The bidimensionality framework of Demaine, Fomin, Hajiaghay, and Thilikos [JACM'05] and the treewidth-pattern-covering approach by Fomin, Lokshtanov, Marx, Pilipczuk, Pilipczuk, and Saurabh [SICOMP'22] give robust tools for designing such algorithms. However, there are still many problems for which we do not know whether subexponential parameterized algorithms exist. The bidimensionality framework is not able to handle weights or directed graphs and the treewidth-pattern-covering approach only works for finding connected solutions. Building on a result by Nederlof [STOC'20], we provide a framework that is able to solve a variety of problems in planar graphs in subexponential parameterized time for which this was previously not known (where the polynomial part of the running time is usually $O(n^{2.49})$). Our framework can handle weights, does not require solutions to contain only few connected components, and applies to cases where the number of potential patterns of a solution is exponential in the parameter. We then use the framework to show that various weighted problems like Weighted Partial Vertex Cover, Maximum-Weight Induced Forest, Minimum-Weight Rooted Simple Minor, and Maximum-Weight Rooted Parallel Induced Minor allow for subexponential parameterized algorithms. This was previously not known for any of them. Moreover, we present a very easy-to-use fragment of our framework. This fragment allows for significantly simpler proofs in the case of Maximum-Weight Independent Set and Maximum $(k, n-k)$-Cut and is able to show a subexponential parameterized algorithm for weighted versions of Densest $k$-Subgraph. Even the unweighted version was not known before and is stated as an open problem in the existing literature.