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Recently, the STEVE-1 approach has been introduced as a method for training generative agents to follow instructions in the form of latent CLIP embeddings. In this work, we present a methodology to extend the control modalities by learning a mapping from new input modalities to the latent goal space of the agent. We apply our approach to the challenging Minecraft domain, and extend the goal conditioning to include the audio modality. The resulting audio-conditioned agent is able to perform on a comparable level to the original text-conditioned and visual-conditioned agents. Specifically, we create an Audio-Video CLIP foundation model for Minecraft and an audio prior network which together map audio samples to the latent goal space of the STEVE-1 policy. Additionally, we highlight the tradeoffs that occur when conditioning on different modalities. Our training code, evaluation code, and Audio-Video CLIP foundation model for Minecraft are made open-source to help foster further research into multi-modal generalist sequential decision-making agents., Comment: Accepted at CoRL 2024: Workshop on Lifelong Learning for Home Robots

We study algebraic shifting of uniform hypergraphs and finite simplicial complexes in the exterior algebra with respect to matrices which are not necessarily generic. Several questions raised by Kalai (2002) are addressed. For instance, it turns out that the combinatorial shifting of Erd\H{o}s$\unicode{x2013}$Ko$\unicode{x2013}$Rado (1961) arises as a special case. Moreover, we identify a sufficient condition for partial shifting to preserve the Betti numbers of a simplicial complex; examples show that this condition is sharp.

A long-standing open question is which graph class is the most general one permitting constant-time constant-factor approximations for dominating sets. The approximation ratio has been bounded by increasingly general parameters such as genus, arboricity, or expansion of the input graph. Amiri and Wiederhake considered $k$-hop domination in graphs of bounded $k$-hop expansion and girth at least $4k+3$; the $k$-hop expansion $f(k)$ of a graph family denotes the maximum ratio of edges to nodes that can be achieved by contracting disjoint subgraphs of radius $k$ and deleting nodes. In this setting, these authors to obtain a simple $O(k)$-round algorithm achieving approximation ratio $\Theta(kf(k))$. In this work, we study the same setting but derive tight bounds: - A $\Theta(kf(k))$-approximation is possible in $k$, but not $k-1$ rounds. - In $3k$ rounds an $O(k+f(k)^{k/(k+1)})$-approximation can be achieved. - No constant-round deterministic algorithm can achieve approximation ratio $o(k+f(k)^{k/(k+1)})$. Our upper bounds hold in the port numbering model with small messages, while the lower bounds apply to local algorithms, i.e., with arbitrary message size and unique identifiers. This means that the constant-time approximation ratio can be \emph{sublinear} in the edge density of the graph, in a graph class which does not allow a constant approximation. This begs the question whether this is an artefact of the restriction to high girth or can be extended to all graphs of $k$-hop expansion $f(k)$., Comment: A shortened version of this article has been accepted for publication at SSS 2024

Constrained Forest Problems (CFPs) as introduced by Goemans and Williamson in 1995 capture a wide range of network design problems with edge subsets as solutions, such as Minimum Spanning Tree, Steiner Forest, and Point-to-Point Connection. While individual CFPs have been studied extensively in individual computational models, a unified approach to solving general CFPs in multiple computational models has been lacking. Against this background, we present the shell-decomposition algorithm, a model-agnostic meta-algorithm that efficiently computes a $(2+\epsilon)$-approximation to CFPs for a broad class of forest functions. To demonstrate the power and flexibility of this result, we instantiate our algorithm for 3 fundamental, NP-hard CFPs in 3 different computational models. For example, for constant $\epsilon$, we obtain the following $(2+\epsilon)$-approximations in the Congest model: 1. For Steiner Forest specified via input components, where each node knows the identifier of one of $k$ disjoint subsets of $V$, we achieve a deterministic $(2+\epsilon)$-approximation in $O(\sqrt{n}+D+k)$ rounds, where $D$ is the hop diameter of the graph. 2. For Steiner Forest specified via symmetric connection requests, where connection requests are issued to pairs of nodes, we leverage randomized equality testing to reduce the running time to $O(\sqrt{n}+D)$, succeeding with high probability. 3. For Point-to-Point Connection, we provide a $(2+\epsilon)$-approximation in $O(\sqrt{n}+D)$ rounds. 4. For Facility Placement and Connection, a relative of non-metric Facility Location, we obtain a $(2+\epsilon)$-approximation in $O(\sqrt{n}+D)$ rounds. We further show how to replace the $\sqrt{n}+D$ term by the complexity of solving Partwise Aggregation, achieving (near-)universal optimality in any setting in which a solution to Partwise Aggregation in near-shortcut-quality time is known., Comment: 42 pages, 4 figures

Minimal free resolutions and minimal free presentations are standard ways to represent the isomorphism type of a persistence module in software. This description is a multi-parameter generalization of the barcode known in one-parameter persistence. Another multi-parameter generalization of this concept is given by fringe presentations (Miller: 2020), which represents the module as the image of a morphism from a free to an injective module. In this paper, we demonstrate a way to calculate the matrix representing a fringe presentation of a given persistence module from a minimal free resolution of it.

With the increasing scale of communication networks, the likelihood of failures grows as well. Since these networks form a critical backbone of our digital society, it is important that they rely on robust routing algorithms which ensure connectivity despite such failures. While most modern communication networks feature robust routing mechanisms, these mechanisms are often fairly complex to design and verify, as they need to account for the effects of failures and rerouting on communication. This paper conceptualizes the design of robust routing mechanisms, with the aim to avoid such complexity. In particular, we showcase \emph{simple} and generic blackbox transformations that increase resilience of routing against independently distributed failures, which allows to simulate the routing scheme on the original network, even in the presence of non-benign node failures (henceforth called faults). This is attractive as the system specification and routing policy can simply be preserved. We present a scheme for constructing such a reinforced network, given an existing (synchronous) network and a routing scheme. We prove that this algorithm comes with small constant overheads, and only requires a minimal amount of additional node and edge resources; in fact, if the failure probability is smaller than $1/n$, the algorithm can come without any overhead at all. At the same time, it allows to tolerate a large number of independent random (node) faults, asymptotically almost surely. We complement our analytical results with simulations on different real-world topologies., Comment: published in IEEE/ACM Transactions on Networking, reposted here to meet archival requirements of grant agencies 15 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1705.04042

Clearing is a simple but effective optimization for the standard algorithm of persistent homology (PH), which dramatically improves the speed and scalability of PH computations for Vietoris--Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris--Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter PH. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter PH that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20., Comment: This is an extended version of a conference paper that appeared at SoCG 2023, see https://drops.dagstuhl.de/opus/volltexte/2023/17865

Gradient clock synchronization (GCS) algorithms minimize the worst-case clock offset between the nodes in a distributed network of diameter $D$ and size $n$. They achieve optimal offsets of $\Theta(\log D)$ locally, i.e. between adjacent nodes as shown by Lenzen et al., and $\Theta(D)$ globally as shown by Biaz and Welch. As demonstrated in the work of Bund et al., this is a highly promising approach for improved clocking schemes for large-scale synchronous Systems-on-Chip (SoC). Unfortunately, in large systems, faults hinder their practical use. State of the art fault-tolerant, as presented by Bund et al., has a drawback that is fatal in this setting: It relies on node and edge replication. For $f=1$, this translates to at least $16$-fold edge replication and high degree nodes, far from the optimum of $2f+1=3$ for tolerating up to $f$ faulty neighbors. In this work, we present a self-stabilizing GCS algorithm for a grid-like directed graph with optimal node in- and out-degrees of $3$ that tolerates $1$ faulty in-neighbor. If nodes fail with independent probability $p\in o(n^{-1/2})$, it achieves asymptotically optimal local skew of $\Theta(\log D)$ with probability $1-o(1)$; this holds under general worst-case assumptions on link delay and clock speed variations, provided they change slowly relative to the speed of the system. The failure probability is the largest possible ensuring that with probabity $1-o(1)$ for each node at most one in-neighbor fails. As modern hardware is clocked at gigahertz speeds and the algorithm can simultaneously sustain a constant number of arbitrary changes due to faults in each clock cycle, this results in sufficient robustness to dramatically increase the size of reliable synchronously clocked SoCs.

Given a discrete-state continuous-time reactive system, like a digital circuit, the classical approach is to first model it as a state transition system and then prove its properties. Our contribution advocates a different approach: to directly operate on the input-output behavior of such systems, without identifying states and their transitions in the first place. We discuss the benefits of this approach at hand of some examples, which demonstrate that it nicely integrates with concepts of self-stabilization and fault-tolerance. We also elaborate on some unexpected artefacts of module composition in our framework, and conclude with some open research questions., Comment: will be published in "Thomas Henzinger Festschrift - Conference celebrating his 60th birthday"

Many critical computing applications rely on secure and dependable time which is reliably synchronized across large distributed systems. Today's time synchronization architectures are commonly based on global navigation satellite systems at the considerable risk of being exposed to outages, malfunction, or attacks against availability and accuracy. This paper describes a practical instantiation of a new global, Byzantine fault-tolerant clock synchronization approach that does not place trust in any single entity and is able to tolerate a fraction of faulty entities while still maintaining synchronization on a global scale among otherwise sovereign network topologies. Leveraging strong resilience and security properties provided by the path-aware SCION networking architecture, the presented design can be implemented as a backward compatible active standby solution for existing time synchronization deployments. Through extensive evaluation, we demonstrate that over 94% of time servers reliably minimize the offset of their local clocks to real-time in the presence of up to 20% malicious nodes, and all time servers remain synchronized with a skew of only 2 ms even after one year of reference clock outage.

Cryptographic signatures can be used to increase the resilience of distributed systems against adversarial attacks, by increasing the number of faulty parties that can be tolerated. While this is well-studied for consensus, it has been underexplored in the context of fault-tolerant clock synchronization, even in fully connected systems. Here, the honest parties of an $n$-node system are required to compute output clocks of small skew (i.e., maximum phase offset) despite local clock rates varying between $1$ and $\vartheta>1$, end-to-end communication delays varying between $d-u$ and $d$, and the interference from malicious parties. So far, it is only known that clock pulses of skew $d$ can be generated with (trivially optimal) resilience of $\lceil n/2\rceil-1$ (PODC `19), improving over the tight bound of $\lceil n/3\rceil-1$ holding without signatures for \emph{any} skew bound (STOC `84, PODC `85). Since typically $d\gg u$ and $\vartheta-1\ll 1$, this is far from the lower bound of $u+(\vartheta-1)d$ that applies even in the fault-free case (IPL `01). We prove matching upper and lower bounds of $\Theta(u+(\vartheta-1)d)$ on the skew for the resilience range from $\lceil n/3\rceil$ to $\lceil n/2\rceil-1$. The algorithm showing the upper bound is, under the assumption that the adversary cannot forge signatures, deterministic. The lower bound holds even if clocks are initially perfectly synchronized, message delays between honest nodes are known, $\vartheta$ is arbitrarily close to one, and the synchronization algorithm is randomized. This has crucial implications for network designers that seek to leverage signatures for providing more robust time. In contrast to the setting without signatures, they must ensure that an attacker cannot easily bypass the lower bound on the delay on links with a faulty endpoint.

In this paper, we refine the (almost) \emph{existentially optimal} distributed Laplacian solver recently developed by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS `21) into an (almost) \emph{universally optimal} distributed Laplacian solver. Specifically, when the topology is known, we show that any Laplacian system on an $n$-node graph with \emph{shortcut quality} $\text{SQ}(G)$ can be solved within $n^{o(1)} \text{SQ}(G) \log(1/\varepsilon)$ rounds, where $\varepsilon$ is the required accuracy. This almost matches our lower bound which guarantees that any correct algorithm on $G$ requires $\widetilde{\Omega}(\text{SQ}(G))$ rounds, even for a crude solution with $\varepsilon \le 1/2$. Even in the unknown-topology case (i.e., standard CONGEST), the same bounds also hold in most networks of interest. Furthermore, conditional on conjectured improvements in state-of-the-art constructions of low-congestion shortcuts, the CONGEST results will match the known-topology ones. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated clique (NCC) model. In this model, we show the existence of a Laplacian solver with round complexity $n^{o(1)} \log(1/\varepsilon)$. The unifying thread of these results, and our main technical contribution, is the study of novel \emph{congested} generalization of the standard \emph{part-wise aggregation} problem. We develop near-optimal algorithms for this primitive in the Supported-CONGEST model, almost-optimal algorithms in (standard) CONGEST, as well as a very simple algorithm for bounded-treewidth graphs with slightly worse bounds. This primitive can be readily used to accelerate the FOCS`21 Laplacian solver. We believe this primitive will find further independent applications.