Achieving agreement among distributed parties is a fundamental task in modern systems, underpinning applications such as consensus in blockchains, coordination in cloud infrastructure, and fault tolerance in critical services. However, this task can be intensive, often requiring a large number of messages to be exchanged as well as many rounds of communication, especially in the presence of Byzantine faults. This makes efficiency a central challenge in the design of practical agreement protocols.

In the online Min-cost Perfect Matching with Delays (\(\textsf{MPMD}\)) problem, \(m\) requests in a metric space are submitted at different times by an adversary. The goal is to match all requests while (i) minimizing the sum of the distances between matched pairs as well as (ii) how long each request remained unmatched after it appeared.

In stable matching, one must find a matching between two sets of agents, commonly men and women, or job applicants and job positions. Each agent has a preference ordering over who they want to be matched with. Moreover a matching is said to be stable if no pair of agents prefer each other over their current matching.

Byzantine Agreement (BA) considers a setting of $n$ parties, out of which up to $t$ can exhibit byzantine (malicious) behavior. Honest parties must decide on a common value (agreement), which must belong to a set determined by the honest inputs (validity). Depending on the use case, this set can grow or shrink, leading to various possible desiderata collectively known as validity conditions. Varying the validity property requirement can affect the regime under which BA is solvable. Our work investigates how the selected validity property impacts BA solvability in the network-agnostic model, where the network can either be synchronous with up to $t_s$ byzantine parties or asynchronous with up to $t_a \leq t_s$ byzantine parties. We give necessary and sufficient conditions for a validity property to render BA solvable, both for the case with cryptographic setup and for the one without. This traces the precise boundary of solvability in the network-agnostic model for every validity property. Our proof of sufficiency provides a universal protocol, that achieves BA for a given validity property whenever the provided conditions are satisfied. We note that, for any non-trivial validity property, the condition $2 \cdot t_s + t_a < n$ is necessary for BA to be solvable, even with cryptographic setup. Specializing this claim to $t_a = 0$ gives that $t < n / 2$ is required whenever one expects a purely synchronous protocol to also work in an asynchronous network when there are no corruptions. This is especially surprising given that, for some validity properties, $t < n$ is a sufficient condition without the last stipulation.

We study the problem of Strong Byzantine Agreement and establish tight upper and lower bounds on communication complexity, parameterized by the actual number of Byzantine faults. Specifically, for a system of n parties tolerating up to t Byzantine faults, out of which only f ≤ t are actually faulty, we obtain the following results: In the partially synchronous setting, we present the first Byzantine Agreement protocol that achieves adaptive communication complexity of 𝒪(n + t ⋅ f) words, which is asymptotically optimal. Our protocol has an optimal resilience of t < n/3. In the asynchronous setting, we prove a lower bound of Ω(n + t²) on the expected number of messages, and design an almost matching protocol with an optimal resilience that solves agreement with 𝒪((n + t²)⋅ log n) words. Our main technical contribution in the asynchronous setting is the utilization of a bipartite expander graph that allows for low-cost information dissemination.

In the Distance-constrained Vehicle Routing Problem (DVRP), we are given a graph with integer edge weights, a depot, a set of n terminals, and a distance constraint D. The goal is to find a minimum number of tours starting and ending at the depot such that those tours together cover all the terminals and the length of each tour is at most D. The DVRP on trees is of independent interest, because it is equivalent to the "virtual machine packing" problem on trees studied by Sindelar et al. [SPAA'11]. We design a simple and natural approximation algorithm for the tree DVRP, parameterized by ε > 0. We show that its approximation ratio is α + ε, where α ≈ 1.691, and in addition, that our analysis is essentially tight. The running time is polynomial in n and D. The approximation ratio improves on the ratio of 2 due to Nagarajan and Ravi [Networks'12]. The main novelty of this paper lies in the analysis of the algorithm. It relies on a reduction from the tree DVRP to the bounded space online bin packing problem via a new notion of "reduced length".