We consider three classification systems for distributed decision tasks: With unbounded computation and certificates, defined by Balliu, D’Angelo, Fraigniaud, and Olivetti [JCSS’18], and with (two flavors of) polynomially bounded local computation and certificates, defined in recent works by Aldema Tshuva and Oshman [OPODIS’23], and by Reiter [PODC’24]. The latter two differ in the way they evaluate the polynomial bounds: the former considers polynomials with respect to the size of the graph, while the latter refers to being polynomial in the size of each node’s local neighborhood. We start by revisiting decision without certificates. For this scenario, we show that the latter two definitions coincide: roughly, a node cannot know the graph size, and thus can only use a running time dependent on its neighborhood. We then consider decision with certificates. With existential certificates ( _1 -type classes), a larger running time defines strictly larger classes of languages: when it grows from being polynomial in each node’s view, through polynomial in the graph’s size, and to unbounded, the derived classes strictly contain each other. With universal certificates ( _1 -type classes), on the other hand, we prove a surprising incomparability result: having running time bounded by the graph size sometimes allows us to decide languages undecidable even with unbounded certificates. We complement these results with other containment and separation results, which together portray a surprisingly complex lattice of strict containment relations between the classes at the base of the three classification systems.

Recent years have seen significant progress in the study of dynamic graph algorithms, and most notably, the introduction of strong lower bound techniques for them (e.g., Henzinger, Krinninger, Nanongkai and Saranurak, STOC 2015; Larsen and Yu, FOCS 2023). As worst-case analysis (adversarial inputs) may lead to the necessity of high running times, a natural question arises: in which cases are high running times really necessary, and in which cases these inputs merely manifest unique pathological cases? Early attempts to tackle this question were made by Nikoletseas, Reif, Spirakis and Yung (ICALP 1995) and by Alberts and Henzinger (Algorithmica 1998), who considered models with very little adversarial control over the inputs, and showed fast algorithms exist for them. The question was then overlooked for decades, until Henzinger, Lincoln and Saha (SODA 2022) recently addressed uniformly random inputs, and presented algorithms and impossibility results for several subgraph counting problems. To tackle the above question more thoroughly, we employ smoothed analysis, a celebrated framework introduced by Spielman and Teng (J. ACM, 2004). An input is proposed by an adversary but then a noisy version of it is processed by the algorithm instead. Parameterized by the amount of adversarial control, this input model fully interpolates between worst-case inputs and a uniformly random input. Doing so, we extend impossibility results for some problems to the smoothed model with only a minor quantitative loss. That is, we show that partially-adversarial inputs suffice to impose high running times for certain problems. In contrast, we show that other problems become easy even with the slightest amount of noise. In addition, we study the interplay between the adversary and the noise, leading to three natural models of smoothed inputs, for which we show a hierarchy of increasing complexity.

A recent breakthrough by Alistarh, Aspnes, Ellen, Gelashvili, and Zhu [STOC 2019] established that there are no extension-based proofs of impossibility for set-agreement within the class of non-uniform iterated immediate snapshot (NIIS) algorithms. An extension-based proof can be modeled as a game between a prover and an algorithm claiming to solve set-agreement, where this algorithm can be from a class 𝒞 of algorithms. Note that the non-existence of extension-based proofs of impossibility for a class 𝒞 of algorithms implies the non-existence of extension-based proofs of impossibility for all classes of algorithms containing 𝒞 . This result has then been revisited by Attiya, Castañeda, and Rajsbaum [OPODIS 2020] who showed that the same holds for the smaller class of uniform iterated immediate snapshot (IIS) algorithms, and even with a slightly stronger prover. The main takeaway message of our work is that these previous results, which show the non-existence of extension-based proofs of impossibility for set-agreement within smaller and smaller universal classes of algorithms, do not necessarily preclude the existence of an extension-based proof of impossibility for set-agreement for an even smaller class of algorithms. To illustrate this, we exhibit an extension-based proof of impossibility for set-agreement for the class of memoryless IIS algorithms. This latter class may not be universal but is strong enough to solve non-trivial tasks such as approximate agreement and renaming. Moreover, we show that the result by Attiya et al. for IIS algorithms, with the stronger prover, does not extend to colorless IIS algorithms. These two results underline the fact that the existence of an extension-based proof of impossibility for a task strongly depends on the considered class of algorithms, and is still open even for set-agreement.

We introduce the certification of solutions to graph problems when access to the input is restricted. This topic has received a lot of attention in the distributed computing setting, and we introduce it here in the context of streaming algorithms, where the input is too large to be stored in memory. Given a graph property , a streaming certification scheme for is a prover-verifier pair where the prover is a computationally unlimited but non-trustable oracle, and the verifier is a streaming algorithm. For any input graph, the prover provides the verifier with a certificate. The verifier then receives the input graph as a stream of edges in an adversarial order, and must check whether the certificate is indeed a proof that the input graph satisfies . The main complexity measure for a streaming certification scheme is its space complexity, defined as the sum of the size of the certificate provided by the oracle, and of the memory space required by the verifier. We give streaming certification schemes for several graph properties, including maximum matching, diameter, degeneracy, and coloring, with space complexity matching the requirement of semi-streaming, i.e., with space complexity O(n n) for n-node graphs. All these problems do not admit semi-streaming algorithms, showing that also in the (semi) streaming setting, certification is sometimes easier than calculation (like NP). For each of these properties, we provide upper and lower bounds on the space complexity of the corresponding certification schemes, many being tight up to logarithmic multiplicative factors. We also show that some graph properties are hard for streaming certification, in the sense that they cannot be certified in semi-streaming, as they require Ω(n^2)-bit certificates.

Distributed certification is a set of mechanisms that allows an all-knowing prover to convince the units of a communication network that the network's state has some desired property, such as being 3-colorable or triangle-free. Classical mechanisms, such as proof labeling schemes (PLS), consist of a message from the prover to each unit, followed by one round of communication between each unit and its neighbors. Later works consider extensions, called distributed interactive proofs, where the prover and the units can have multiple rounds of communication before the communication among the units. Recently, Bick, Kol, and Oshman (SODA '22) defined a zero-knowledge version of distributed interactive proofs, where the prover convinces the units of the network's state without revealing any other information about the network's state or structure. In their work, they propose different variants of this model and show that many graph properties of interest can be certified with them. In this work, we define and study distributed non-interactive zero-knowledge proofs (dNIZK); these can be seen as a non-interactive version of the aforementioned model, and also as a zero-knowledge version of PLS. We prove the following: - There exists a dNIZK protocol for 3-coloring with O(log n)-bit messages from the prover and O(log n)-size messages among neighbors. - There exists a family of dNIZK protocols for triangle-freeness, that presents a trade-off between the size of the messages from the prover and the size of the messages among neighbors. - There exists a dNIZK protocol for any graph property in NP in the random oracle models, which is secure against an arbitrary number of malicious parties.

We study two fundamental problems of distributed computing, consensus and approximate agreement, through a novel approach for proving lower bounds and impossibility results, that we call the asynchronous speedup theorem. For a given n-process task Ρ and a given computational model M, we define a new task, called the closure of Ρ with respect to M. The asynchronous speedup theorem states that if a task Ρ is solvable in t ≥ 1 rounds in M, then its closure w.r.t. M is solvable in t ≥ 1 rounds in M. We prove this theorem for iterated models, as long as the model allows solo executions. We illustrate the power of our asynchronous speedup theorem by providing a new proof of the wait-free impossibility of consensus using read/write registers, and a new proof of the wait-free impossibility of solving consensus using registers and test&set objects for > 2. The proof is merely by showing that, in each case, the closure of consensus (w.r.t. the corresponding model) is consensus itself. Our main application is the study of the power of additional objects, namely test&set and binary consensus, for wait-free solving approximate agreement faster. By analyzing the closure of approximate agreement w.r.t. each of the two models, we show that while these objects are more powerful than read/write registers from the computability perspective, they are not more powerful as far as helping solving approximate agreement faster is concerned.

Given a positive integer k, k-set agreement is the distributed task in which each process i∈ [n] in a group of n processing nodes starts with an input value x_i in the set {0,…,k}, and must output a value y_i such that (1) for every i ∈ [n], y_i is the input value of some process, and (2)|{y_i : i∈ [n]}|≤ k. That is, at most k different values in total must be outputted by the processes. The case k=1 correspond to (binary) consensus, arguably the most studied problem in distributed computing. While lower bounds for consensus have been obtained for most of the standard distributed computing models, the design of lower bounds for k-set agreement with k>1 is notoriously known to be much more difficult, and remains open for many models. The main techniques for designing lower bounds for k-set agreement with k>1 use tools from algebraic topology. The algebraic topology tools are difficult to manipulate, and require a lot of care for avoiding mistakes. This difficulty increases when the communications are mediated by a network of arbitrary structure. Recently, the KNOWALL model has been specifically designed as a first attempt to understand the LOCAL model through the lens of algebraic topology, and Castañeda et al.(2021) have designed lower bounds for k-set agreement in the KNOWALL model, with applications to dynamic networks. In this work, we re-prove the same lower bound for k-set agreement in the KNOWALL model. This new proof stands out in its simplicity, which makes it accessible to a broader audience, and increases confidence in the result.

We develop a new lower bound for k-set agreement in synchronous message-passing systems connected by an arbitrary directed communication network, where up to t processes may crash. Our result thus generalizes the [t/k] + 1 lower bound for complete networks in the t-resilient model by Chaudhuri, Herlihy, Lynch, and Tuttle [JACM 2000]. Moreover, it generalizes two lower bounds for oblivious algorithms in synchronous systems connected by an arbitrary undirected communication network known to the processes, namely, the domination number-based lower bound by Castaneda, Fraigniaud, Paz, Rajsbaum, Roy, and Travers [TCS 2021] for failure-free processes, and the radius-based lower bound in the t-resilient model by Fraigniaud, Nguyen, and Paz [STACS 2024]. Our topological proof non-trivially generalizes and extends the connectivity-based approach for the complete network, as presented in the book by Herlihy, Kozlov, and Rajsbaum (2013). It is based on a sequence of shellable carrier maps that, starting from a shellable input complex, determine the evolution of the protocol complex: During the first [t/k] rounds, carrier maps that crash exactly k processes per round are used, which ensure high connectivity of their images. A Sperner's lemma style argument can thus be used to prove that k-set agreement is still impossible by that round. From round [t/k] + 1 up to our actual lower bound, a novel carrier map is employed, which maintains high connectivity. As a by-product, our proof also provides a strikingly simple lower-bound for k-set agreement in synchronous systems with an arbitrary communication network, where exactly t >= 0 processes crash initially, i.e., before taking any step. We demonstrate that the resulting additional agreement overhead can be expressed via an appropriately defined radius of the communication graphs, and show that the usual input pseudosphere complex for k-set agreement can be replaced by an exponentially smaller input complex based on Kuhn triangulations, which we prove to be also shellable. 2012 ACM Subject Classification Theory of computation -> Distributed computing models

Consensus is arguably the most studied problem in distributed computing as a whole, and particularly in the distributed message-passing setting. In this latter framework, research on consensus has considered various hypotheses regarding the failure types, the memory constraints, the algorithmic performances (e.g., early stopping and obliviousness), etc. Surprisingly, almost all of this work assumes that messages are passed in a complete network, i.e., each process has a direct link to every other process. A noticeable exception is the recent work of Castaneda et al. (Inf. Comput. 2023) who designed a generic oblivious algorithm for consensus running in radius(G, t) rounds in every graph G, when up to t nodes can crash by irrevocably stopping, where t is smaller than the node-connectivity kappa of G. Here, radius(G, t) denotes a graph parameter called the radius of G whenever up to t nodes can crash. For t = 0, this parameter coincides with radius(G), the standard radius of a graph, and, for G = K-n, the running time radius(K-n, t) = t + 1 of the algorithm exactly matches the known round-complexity of consensus in the clique K-n. Our main result is a proof that radius(G, t) rounds are necessary for oblivious algorithms solving consensus in G when up to t nodes can crash, thus validating a conjecture of Castaneda et al., and demonstrating that their consensus algorithm is optimal for any graph G. We also extend the result of Castaneda et al. to two different settings: First, to the case where the number t of failures is not necessarily smaller than the connectivity kappa of the considered graph; Second, to the k-set agreement problem for which agreement is not restricted to be on a single value as in consensus, but on up to k different values.

A key result of distributed computing in asynchronous systems is a characterization for the wait-free solvability of colorless tasks by the existence of a continuous map from the task's input complex (representing the valid input configurations) to its output complex (representing the valid output configurations) which respects that task's specification. This natural characterization led to many proofs, mainly of impossibility: showing that a colorless task is not wait-free solvable, can be done by proving that there is no continuous map (respecting the task's specification) between two simplicial complexes, which can be done using classical topological machinery. The seminal work of Herlihy and Shavit (JACM '99) characterized the solvability of general (not necessarily colorless) tasks. However, this characterization is much more involved than the colorless one, as it uses the new notions of chromatic subdivisions and color-preserving maps. The characterization asks whether there exists a chromatic subdivision of the input complex and a colorpreserving map from the resulting subdivided complex to the output complex. This characterization is much harder to check as there are no ready-made topological tools for it, and in fact, finding such a subdivision and map is related to finding an algorithm for the task. This work presents a new and simpler characterization for the solvability of general tasks with three processes. In our characterization, the output complex undergoes a bounded number of simple combinatorial transformations. Then, we check if there is a continuous map from the input complex to the resulting output complex; we show that this suffices for determining whether the original task is solvable. Our approach provides a new and more direct way for deciding the solvability of a task, and also for proving impossibility and possibility of wait-free solutions for general tasks.

Modeling distributed computing in a way enabling the use of formal methods is a challenge that has been approached from different angles, among which two techniques emerged at the turn of the century: protocol complexes, and directed algebraic topology. In both cases, the considered computational model generally assumes communication via shared objects, typically a shared memory consisting of a collection of read-write registers. Our paper is concerned with network computing, where the processes are located at the nodes of a network, and communicate by exchanging messages along the edges of that network. Applying the topological approach for verification in network computing is a considerable challenge, mainly because the presence of identifiers assigned to the nodes yields protocol complexes whose size grows exponentially with the size of the underlying network. However, many of the problems studied in this context are of local nature, and their definitions do not depend on the identifiers or on the size of the network. We leverage this independence in order to meet the above challenge, and present $\textit{local}$ protocol complexes, whose sizes do not depend on the size of the network. As an application of the design of "compact" protocol complexes, we reformulate the celebrated lower bound of $\Omega(\log^*n)$ rounds for 3-coloring the $n$-node ring, in the algebraic topology framework.

Guess Who is a two-player search game in which each player chooses a character from a deck of 24 cards, and has to infer the other player's character by asking yes-no questions. A simple binary search strategy allows the starting player find the opponent's character by asking 5 questions only, when the opponent is honest.Real-life observations show that in more realistic scenarios, the game is played against adversaries that do not strictly follow the rules, e.g., kids. Such players might decide to answer all questions at once, answer only part of the questions as they do not know the answers to all, and even lie occasionally. We devise strategies for such scenarios using techniques from error-correcting and erasure codes. This connects to a recent line of work on search problems on graphs and trees with unreliable auxiliary information, and could be of independent interest.

We study the problem of solving consensus in synchronous directed dynamic networks, in which communication is controlled by an oblivious message adversary that picks the communication graph to be used in a round from a fixed set of graphs D arbitrarily. In this fundamental model, determining consensus solvability and designing efficient consensus algorithms is surprisingly difficult. Enabled by a decision procedure that is derived from a well-established previous consensus solvability characterization for a given set D , we study, for the first time, the time complexity of solving consensus in this model: We provide both upper and lower bounds for this time complexity, and also relate it to the number of iterations required by the decision procedure. Among other results, we find that reaching consensus under an oblivious message adversary can take exponentially longer than both deciding consensus solvability and broadcasting the input value of some unknown process to all other processes.

The k -center problem is a central optimization problem with numerous applications for machine learning, data mining, and communication networks. Despite extensive study in various scenarios, it surprisingly has not been thoroughly explored in the traditional distributed setting, where the communication graph of a network also defines the distance metric. We initiate the study of the k -center problem in a setting where the underlying metric is the graph’s shortest path metric in three canonical distributed settings: the local, congest, and clique models. Our results encompass constant-factor approximation algorithms and lower bounds in these models, as well as hardness results for the bi-criteria approximation setting.

The topological view on distributed computing represents a task T as a relation Delta between the complex I of its inputs and the complex O of its outputs. A cornerstone result in the field is an elegant computability characterization of the solvability of colorless tasks in terms of I, O and Delta. Essentially, a colorless task is wait-free solvable if and only if there is a continuous map from the geometric realization of I to that of O that respects Delta. This paper makes headway towards providing an analogous characterization for general tasks, which are not necessarily colorless, by concentrating on the case of three-process inputless tasks. Our key contribution is identifying local articulation points as an obstacle for the solvability of general tasks, and defining a topological deformation on the output complex of a task T, which eliminates these points by splitting them, to obtain a new task T', with an adjusted relation Delta' between the input complex I and an output complex O' without articulation points. We obtain a new characterization of wait-free solvability of three-process general tasks: T is wait-free solvable if and only if there is a continuous map from the geometric realization of I to that of O' that respects Delta'.

The paper compares two generic techniques for deriving lower bounds and impossibility results in distributed computing. First, we prove a speedup theorem (a-la Brandt, 2019), for wait-free colorless algorithms, aiming at capturing the essence of the seminal round-reduction proof establishing a lower bound on the number of rounds for 3-coloring a cycle (Linial, 1992), and going by backward induction. Second, we consider FLP-style proofs, aiming at capturing the essence of the seminal consensus impossibility proof (Fischer, Lynch, and Paterson, 1985) and using forward induction. We show that despite their very different natures, these two forms of proof are tightly connected. In particular, we show that for every colorless task $\Pi$, if there is a round-reduction proof establishing the impossibility of solving $\Pi$ using wait-free colorless algorithms, then there is an FLP-style proof establishing the same impossibility. For 1-dimensional colorless tasks (for an arbitrary number $n\geq 2$ of processes), we prove that the two proof techniques have exactly the same power, and more importantly, both are complete: if a 1-dimensional colorless task is not wait-free solvable by $n\geq 2$ processes, then the impossibility can be proved by both proof techniques. Moreover, a round-reduction proof can be automatically derived, and an FLP-style proof can be automatically generated from it. Finally, we illustrate the use of these two techniques by establishing the impossibility of solving any colorless covering task of arbitrary dimension by wait-free algorithms.

Consensus is one of the most fundamental problems in distributed computing. This paper studies the consensus problem in a synchronous dynamic directed network, in which communication is controlled by an oblivious message adversary. The question when consensus is possible in this model has already been studied thoroughly in the literature from a combinatorial perspective, and is known to be challenging. This paper presents a topological perspective on consensus solvability under oblivious message adversaries, which provides interesting new insights. Our main contribution is a topological characterization of consensus solvability, which also leads to explicit decision procedures. Our approach is based on the novel notion of a communication pseudosphere, which can be seen as the message-passing analog of the well-known standard chromatic subdivision for wait-free shared memory systems. We further push the elegance and expressiveness of the "geometric" reasoning enabled by the topological approach by dealing with uninterpreted complexes, which considerably reduce the size of the protocol complex, and by labeling facets with information flow arrows, which give an intuitive meaning to the implicit epistemic status of the faces in a protocol complex.

The sinkless orientation problem plays a key role in understanding the foundations of distributed computing. The problem can be used to separate two fundamental models of distributed graph algorithms, LOCAL and SLOCAL: the locality of sinkless orientation is $\Omega(\log n)$ in the deterministic LOCAL model and $O(\log \log n)$ in the deterministic SLOCAL model. Both of these results are known by prior work, but here we give new simple, self-contained proofs for them.

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and ($s,t$)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an $m$-edge graph, there exists no dynamic algorithms with both $O(m^{1/2 - \epsilon})$ update time and $ O(m^{1 -\epsilon})$ query time, for any small $\epsilon > 0$. Note that for ($s,t$)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and $O(m)$ query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching.