The question of "what can be computed locally?" lies at the heart of distributed computing in networks. As established in Naor and Stockmeyer's seminal paper (STOC 1993, Edsger W. Dijkstra Prize in Distributed Computing 2025), this question is undecidable, even for graph problems whose solutions can be checked locally. In this paper, we adopt a novel perspective on the question, by asking for which classes Π of problems, and for which classes G of graphs, all problems in Π can be solved efficiently in a distributed manner in all graphs of G. This paper focuses on two natural candidates for such an approach, namely the class of problems expressible in first-order logic (FO), because they possess an intrinsic form of locality thanks to Gaifman's theorem, and the class of graphs with bounded expansion, because they form a large class of graphs encompassing, e.g., planar, bounded-genus, bounded-treewidth, and bounded-degree graphs, as well as graphs excluding a fixed minor or topological minor, sparse Erdös--Rényi graphs (a.a.s.), and several network models such as stochastic block models for suitable parameter ranges.

Abstract We establish that every monadic second-order logic (MSO) formula on graphs with bounded treedepth is decidable in a constant number of rounds within the model. To our knowledge, this marks the first meta-theorem regarding distributed model checking. Various optimization problems on graphs are expressible in MSO. Examples include determining whether a graph G has a clique of size k , whether it admits a coloring with k colors, whether it contains a graph H as a subgraph or minor, or whether terminal vertices in G could be connected via vertex-disjoint paths. Our meta-theorem significantly enhances the work of Bousquet et al. (in: 41st ACM Symposium on Principles of Distributed Computing (PODC), 2022), which was focused on distributed certification of MSO on graphs with bounded treedepth. Moreover, our results can be extended to solving optimization and counting problems expressible in MSO, in graphs of bounded treedepth.

We establish that for every first-order logic (FO) formula ϕ, which captures a vast number of computational problems on graphs, and every graph class G of bounded expansion, there exists a deterministic distributed algorithm that, for any n-node graph G ∈ G with diameter D, determines whether G ⊨ ϕ within O(D+log n) rounds in the standard CONGEST model. Graph classes of bounded expansion encompass many well-known families of sparse graphs, including planar graphs, bounded-genus graphs, bounded-treedepth graphs, bounded-treewidth graphs, bounded-degree graphs, graphs that exclude a fixed graph H as a minor or topological minor, random graphs with constant average degree (a.a.s.), and many network models (e.g., stochastic block models) for some ranges of parameters.

One of the most basic techniques in algorithm design consists of breaking a problem into subproblems and then proceeding recursively. In the case of graph algorithms, one way to implement this approach is through separator sets. Given a graph G = (V, E), a subset of nodes S ⊆ V is called a separator set of G if the size of each connected component of G - S is at most 2/3 · |V|. The most useful separator sets are those that satisfy certain restrictions of cardinality or structure.

ABSTRACT In the minimum clique routing problem on cycles mcrpc, we are given a cycle together with a set of demands (weighted terminals pairs) and the goal is to route all the pairs minimizing the maximum weight clique of the intersection graph induced by the routing. The nodes of this graph are the demands with their corresponding weights and two demands are adjacent when their routes share at least one arc. In this work, we are not only interested in the mcrpc but also in two natural subproblems. First, we consider the situation where the demands are disjoint, in the sense that every two demands do not share any of their corresponding terminals. Second, we analyze the subproblem where the weights of the routes are all equal. We first show that the problem is NP‐hard even in the subproblem of disjoint demands. For the case of arbitrary weights, we exhibit a simple combinatorial 2‐approximation algorithm and a ‐approximation algorithm based on rounding a solution of a relaxation of an integer linear programming formulation of our problem. Finally, we give a fixed parameter tractable algorithm for the case of uniform weights, whose parameter is the maximum number of demands for which a demand exists whose terminals alternate in the cycle with the terminals of each of them.

We establish that every monadic second-order logic (MSO) formula on graphs with bounded treedepth is decidable in a constant number of rounds within the CONGEST model. To our knowledge, this marks the first meta-theorem regarding distributed model-checking. Various optimization problems on graphs are expressible in MSO. Examples include determining whether a graph G has a clique of size k, whether it admits a coloring with k colors, whether it contains a graph H as a subgraph or minor, or whether terminal vertices in G could be connected via vertex-disjoint paths. Our meta-theorem significantly enhances the work of Bousquet et al. [PODC 2022], which was focused on distributed certification of MSO on graphs with bounded treedepth. Moreover, our results can be extended to solving optimization and counting problems expressible in MSO, in graphs of bounded treedepth.

We establish that every monadic second-order logic (MSO) formula on graphs with bounded treedepth is decidable in a constant number of rounds within the CONGEST model. To our knowledge, this marks the first meta-theorem regarding distributed model-checking. Various optimization problems on graphs are expressible in MSO. Examples include determining whether a graph $G$ has a clique of size $k$, whether it admits a coloring with $k$ colors, whether it contains a graph $H$ as a subgraph or minor, or whether terminal vertices in $G$ could be connected via vertex-disjoint paths. Our meta-theorem significantly enhances the work of Bousquet et al. [PODC 2022], which was focused on distributed certification of MSO on graphs with bounded treedepth. Moreover, our results can be extended to solving optimization and counting problems expressible in MSO, in graphs of bounded treedepth.

A proof-labeling scheme (PLS) for a boolean predicate Π on labeled graphs is a mechanism used for certifying the legality with respect to Π of global network states in a distributed manner. In a PLS, a certificate is assigned to each processing node of the network, and the nodes are in charge of checking that the collection of certificates forms a global proof that the system is in a correct state, by exchanging the certificates once, between neighbors only. The main measure of complexity is the size of the certificates. Many PLSs have been designed for certifying specific predicates, including cycle-freeness, minimum-weight spanning tree, planarity, etc. In 2021, a breakthrough has been obtained, as a "meta-theorem" stating that a large set of properties have compact PLSs in a large class of networks. Namely, for every MSO₂ property Π on labeled graphs, there exists a PLS for Π with O(log n)-bit certificates for all graphs of bounded tree-depth. This result has been extended to the larger class of graphs with bounded tree-width, using certificates on O(log² n) bits. We extend this result even further, to the larger class of graphs with bounded clique-width, which, as opposed to the other two aforementioned classes, includes dense graphs. We show that, for every MSO₁ property Π on labeled graphs, there exists a PLS for Π with O(log² n)-bit certificates for all graphs of bounded clique-width. As a consequence, certifying families of graphs such as distance-hereditary graphs and (induced) P₄-free graphs (a.k.a., cographs) can be done using a PLS with O(log² n)-bit certificates, merely because each of these two classes can be specified in MSO₁. In fact, we show that certifying P₄-free graphs can be done with certificates on O(log n) bits only. This is in contrast to the class of C₄-free graphs (which does not have bounded clique-width) which requires Ω̃(√n)-bit certificates.

During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of the relative powers of these models. That is, for every pair of such models, we determine whether one is (strictly) stronger than the other, or whether the two models are incomparable. The separation results are obtained by approaching communication complexity through an original angle, which may be of an independent interest. The two players are not bounded to compute the value of a binary function, but the combined outputs of the two players are constrained by this value. In particular, we introduce the XOR-Index problem, in which Alice is given a binary vector x ∈ {0,1}ⁿ together with an index i ∈ [n], Bob is given a binary vector y ∈ {0,1}ⁿ together with an index j ∈ [n], and, after a single round of 2-way communication, Alice must output a boolean out_A, and Bob must output a boolean out_B, such that out_A ∧ out_B = x_j⊕ y_i. We show that the communication complexity of XOR-Index is Ω(n) bits.

During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of the relative powers of these models. That is, for every pair of such models, we determine whether one is (strictly) stronger than the other, or whether the two models are incomparable.

Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACM-SIAM symposium on discrete algorithms (SODA), pp 1096–115, 2020. https://doi.org/10.1137/1.9781611975994.67 ) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM protocol), and uses small certificates, on $$O(\log n)$$ bits in n-node networks. We show that a single interaction with the prover suffices, and randomization is unecessary, by providing an explicit description of a proof-labeling scheme for planarity, still using certificates on just $$O(\log n)$$ bits. We also show that there are no proof-labeling schemes—in fact, even no locally checkable proofs—for planarity using certificates on $$o(\log n)$$ bits.

In distributed interactive proofs, the nodes of a graph G interact with a powerful but untrustable prover who tries to convince them, in a small number of rounds and through short messages, that G satisfies some property. This series of interactions is followed by a phase of distributed verification, which may be either deterministic or randomized, where nodes exchange messages with their neighbors. The nature of this last verification round defines the two types of interactive protocols. We say that the protocol is of Arthur-Merlin type if the verification round is deterministic. We say that the protocol is of Merlin-Arthur type if, in the verification round, the nodes are allowed to use a fresh set of random bits. In the original model introduced by Kol, Oshman, and Saxena [PODC 2018], the randomness was private in the sense that each node had only access to an individual source of random coins. Crescenzi, Fraigniaud, and Paz [DISC 2019] initiated the study of the impact of shared randomness (the situation where the coin tosses are visible to all nodes) in the distributed interactive model. In this work, we continue that research line by showing that the impact of the two forms of randomness is very different depending on whether we are considering Arthur-Merlin protocols or Merlin-Arthur protocols. While private randomness gives more power to the first type of protocols, shared randomness provides more power to the second. Our results also connect shared randomness in distributed interactive proofs with distributed verification, and new lower bounds are obtained.

Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM protocol), and uses small certificates, on O(log n) bits in n-node networks. We show that a single interaction from the prover suffices, and randomization is unecessary, by providing an explicit description of a proof-labeling scheme for planarity, still using certificates on just O(log n) bits. We also show that there are no proof-labeling schemes --- in fact, even no locally checkable proofs --- for planarity using certificates on o(log n) bits.

The broadcast congested clique model (BClique) is a message-passing model of distributed computation where $n$ nodes communicate with each other in synchronous rounds. First, in this paper we prove that there is a one-round, deterministic algorithm that reconstructs the input graph $G$ if the graph is $d$-degenerate, and rejects otherwise, using bandwidth $b=\mathcal{O}(d \cdot \log n)$. Then, we introduce a new parameter to the model. We study the situation where the nodes, initially, instead of knowing their immediate neighbors, know their neighborhood up to a fixed radius $r$. In this new framework, denoted ${{\sc BClique}}[r]$, we study the problem of detecting, in $G$, an induced cycle of length at most $k$ (${\sc Cycle}_{\leq k}$) and the problem of detecting an induced cycle of length at least $k+1$ (${\sc Cycle}_{>k}$). We give upper and lower bounds. We show that if each node is allowed to see up to distance $r={\lfloor k/2 \rfloor + 1}$, then a polylogarithmic bandwidth is sufficient for solving ${\sc Cycle}_{>k}$ with only two rounds. Nevertheless, if nodes were allowed to see up to distance $r=\lfloor k/3 \rfloor$, then any one-round algorithm that solves ${\sc Cycle}_{>k}$ needs the bandwidth $b$ to be at least $\Omega(n/\log n)$. We also show the existence of a one-round, deterministic ${{\sc BClique}}$ algorithm that solves ${\sc Cycle}_{\leq k}$ with bandwitdh $b=\mathcal{O}(n^{1/\lfloor{k/2}\rfloor} \cdot \log n)$. On the negative side, we prove that, if $\epsilon \leq 1/3$ and $0 < r \leq k/4 $, then any $\epsilon$-error, $R$-round, $b$-bandwidth algorithm in the ${{\sc BClique}}[r]$ model that solves problem ${\sc Cycle}_{\leq k}$ satisfies $R \cdot b = \Omega(n^{1/\lfloor{k/2}\rfloor})$.

In this paper we present distributed testing algorithms of graph properties in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error testing algorithms in the general graph model. We first describe a general procedure for converting $ε$-testers with a number of rounds $f(D)$, where $D$ denotes the diameter of the graph, to $O((\log n)/ε)+f((\log n)/ε)$ rounds, where $n$ is the number of processors of the network. We then apply this procedure to obtain an optimal tester, in terms of $n$, for testing bipartiteness, whose round complexity is $O(ε^{-1}\log n)$, which improves over the $poly(ε^{-1} \log n)$-round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property. In the second part of the paper we design algorithms for testing whether the network is $H$-free for any connected $H$ of size up to four with round complexity of $O(ε^{-1})$. This improves over the $O(ε^{-2})$-round algorithms for testing triangle freeness by Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size $4$ by Fraigniaud et al. (DISC 2016). In the last part we generalize the global tester by Iwama and Yoshida (ITCS 2014) of testing $k$-path freeness to testing the exclusion of any tree of order $k$. We then show how to simulate this algorithm in the CONGEST-model in $O(k^{k^2+1}\cdotε^{-k})$ rounds.

In the broadcast version of the congested clique model, nodes communicate in synchronous rounds by writing ‐bit messages on a whiteboard, which is visible to all of them. The joint input to the nodes is an undirected ‐node graph , with node receiving the list of its neighbors in . Our goal is to design a protocol at the end of which the information contained in the whiteboard is enough for reconstructing . It has already been shown that there is a one‐round protocol for reconstructing graphs with bounded degeneracy. The main drawback of that protocol is that the degeneracy of the input graph must be knowna prioriby the nodes. Moreover, the protocol fails when applied to graphs with degeneracy larger than . In this article, we address this issue by looking forrobustreconstruction protocols, that is, protocols which always give the correct answer and work efficiently when the input is restricted to a certain class. We introduce a very simple, two‐round protocol that we call Robust‐Reconstruction. We prove that this protocol is robust for reconstructing the class of Barabási‐Albert trees with (expected) message size . Moreover, we present computational evidence suggesting that Robust‐Reconstructionalso generates logarithmic size messages for arbitrary Barabási‐Albert networks. Finally, we stress the importance of the preferential attachment mechanism (used in the construction of Barabási‐Albert networks) by proving that Robust‐Reconstructiondoes notgenerate short messages for random recursive trees. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(1), 82–91 2016

The CONGEST model is a synchronous, message-passing model of distributed computation in which each node can send (possibly different) messages of O(log n) bits along each of its incident communication links in each round, where n is the number of computing nodes in the system. In the particular case where the communication network is a complete graph, we have the unicast congested clique model. On the other end is the broadcast version of the congested clique model, in which each node can only broadcast a single message over all its links in each round. In this paper we explore the space, in terms of round complexity, that lies between these two congested clique models. Hence, we parametrize the congested clique model with the range r, the maximum number of different messages a node can send over its incident links in one round. Additionally, we study the effect of the bandwidth b, the maximum size in bits of these messages. We show that the space between the unicast and broadcast congested clique models is very rich and interesting. For instance, we show that a problem (especially designed for this work) takes Ω(n/ log n) rounds in the broadcast model (r = 1), while it can be solved in two rounds if two messages can be sent (r = 2). Other gaps are found in other parts of the spectrum of values of r. We do this by providing techniques to simulate protocols with different parameters. Therefore, we conclude that, with respect to their power to solve certain problems, there is a strict hierarchy of congested clique models.

In this paper we study distributed algorithms on massive graphs where links represent a particular relationship between nodes (for instance, nodes may represent phone numbers and links may indicate telephone calls). Since such graphs are massive they need to be processed in a distributed and streaming way. When computing graph theoretic properties, nodes become natural units for distributed computation. Links do not necessarily represent communication channels between the computing units and therefore do not restrict the communication flow. Our goal is to model and analyze the computational power of such distributed systems where one computing unit is assigned to each node. Communication takes place on a whiteboard where each node is allowed to write at most one message. Every node can read the contents of the whiteboard and, when activated, can write one small message based on its local knowledge. When the protocol terminates its output is computed from the final contents of the whiteboard. We describe four synchronization models for accessing the whiteboard. We show that message size and synchronization power constitute two orthogonal hierarchies for these systems. We exhibit problems that {\it separate} these models, i.e., that can be solved in one model but not in a weaker one, even with increased message size. These problems are related to maximal independent set and connectivity. We also exhibit problems that require a given message size independently of the synchronization model.

In this paper we ask which properties of a distributed network can be computed from a few amount of local information provided by its nodes. The distributed model we consider is a restriction of the classical CONGEST (distributed) model and it is close to the simultaneous messages (communication complexity) model defined by Babai, Kimmel and Lokam. More precisely, each of these n nodes-which only knows its own ID and the IDs of its neighbors- is allowed to send a message of O(log n) bits to some central entity, called the referee. Is it possible for the referee to decide some basic structural properties of the network topology G? We show that simple questions like, "does G contain a square?", "does G contain a triangle?" or "Is the diameter of G at most 3?" cannot be solved in general. On the other hand, the referee can decode the messages in order to have full knowledge of G when G belongs to many graph classes such as planar graphs, bounded tree width graphs and, more generally, bounded degeneracy graphs. We leave open questions related to the connectivity of arbitrary graphs.

We investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter never changes its state. This process is a particular case of bootstrap percolation. We show that in any cubic graph, with high probability, the information will not spread to all vertices in the graph if $p<\frac{1}{2}$ . We give families of graphs in which information spreads to all vertices with high probability for relatively small values of p.

BACKGROUND: To allow the survival of the population in the absence of nitrogen, some cyanobacteria strains have developed the capability of differentiating into nitrogen fixing cells, forming a characteristic pattern. In this paper, the process by which cyanobacteria differentiates from vegetative cells into heterocysts in the absence of nitrogen and the elements of the gene network involved that allow the formation of such a pattern are investigated. METHODS: A simple gene network model, which represents the complexity of the differentiation process, and the role of all variables involved in this cellular process is proposed. Specific characteristics and details of the system's behavior such as transcript profiles for ntcA, hetR and patS between consecutive heterocysts were studied. RESULTS: The proposed model is able to capture one of the most distinctive features of this system: a characteristic distance of 10 cells between two heterocysts, with a small standard deviation according to experimental variability. The system's response to knock-out and over-expression of patS and hetR was simulated in order to validate the proposed model against experimental observations. In all cases, simulations show good agreement with reported experimental results. CONCLUSION: A simple evolution mathematical model based on the gene network involved in heterocyst differentiation was proposed. The behavior of the biological system naturally emerges from the network and the model is able to capture the spacing pattern observed in heterocyst differentiation, as well as the effect of external perturbations such as nitrogen deprivation, gene knock-out and over-expression without specific parameter fitting.