Algorithmic meta-theorems, stating that graph properties expressible in some particular logic can be decided efficiently in graph classes having some specific structural properties, are now standard in sequential graph algorithms. One of the most classic examples is Courcelle's theorem: all properties expressible in Monadic Second-Order logic (MSO) are decidable in linear time in graphs of bounded treewidth.

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.

In this article we investigate the computational complexity of predicting two dimensional freezing majority cellular automata with states { − 1 , + 1 } , where the local interactions are based on an L-shaped neighborhood structure. In these automata, once a cell reaches state + 1 , it remains fixed in that state forever, while cells in state − 1 update to the most represented state among their neighborhoods. We consider L-shaped neighborhoods, which mean that the vicinity of a given cell c consists in a subset of cells in the north and east of c . We focus on the prediction problem, a decision problem that involves determining the state of a given cell after a given number of time-steps. We prove that when restricted to the simplest L-shaped neighborhood, consisting of the central cell and its nearest north and east neighbors, the prediction problem belongs to NC , meaning it can be solved efficiently in parallel. We generalize this result for any L-shaped neighborhood of size two. On the other hand, for other L-shaped neighborhoods, the problem becomes P -Complete, indicating that the problem might be inherently sequential.

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.

We study the problem of certifying whether a graph is k-colorable with a locally checkable proof (LCP) that is able to hide the k-coloring from the verifier, in the sense that no algorithm can (completely) extract a k-coloring from the certificate. Motivated by the search for promise-free separations of extensions of the LOCAL model in the context of locally checkable labeling (LCL) problems, we also require the LCPs to satisfy what we call the strong soundness property. We focus on the case of 2-coloring and show that strong and hiding LCPs for 2-coloring exist in specific graph classes and require only O (log n)-sized certificates. Furthermore, when the input is promised to be a cycle or contains a node of degree 1, we show the existence of strong and hiding LCPs even in an anonymous network and with constant-size certificates. Despite these upper bounds, we prove that there are no strong and hiding LCPs for 2-coloring in general, regardless of certificate size. Along the way, we give a characterization of the hiding property for the general k-coloring problem that appears to be a key component for future investigations in this context.

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.

In this work, we provide a procedure that allows us to transform certain kinds of deterministic Boolean networks on minterm or maxterm functions into symmetric ones, so inferring that such symmetrizable networks can present only periodic points of periods 1 or 2. In particular, we deal with generalized parallel (or synchronous) dynamical systems (GPDS) over undirected graphs, i.e., discrete parallel dynamical systems over undirected graphs where some of the self-loops may not appear. We also study the class of anti-symmetric GPDS (which are non-symmetrizable), proving that their periodic orbits have period 4. In addition, we introduce a class of non-symmetrizable systems which admit periodic orbits with arbitrary large periods.

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.

Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. However, for many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded Π- Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties Π, even when parameterized by k + t . We show that the situation changes when we are looking for a connected induced subgraph satisfying Π. In particular, we show that the Connected Secluded Π -Subgraph problem is FPT when parameterized by just t for many important graph properties Π.

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.

We present deterministic constant-round protocols for the graph connectivity problem in the model where each of the n nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Specifically, we exhibit a deterministic protocol that computes the connected components of the input graph in [1/ε] rounds, each player communicating O(nε ⋅ log n) bits per round, with 0 < ε ≤ 1.