Becker, F., Kosowski, A., Matamala, M., Nisse, N., Rapaport, I., Suchan, K., et al. (2015). Allowing each node to communicate only once in a distributed system: shared whiteboard models. Distrib. Comput., 28(3), 189–200.
Abstract: 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 way. When computing graphtheoretic 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 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.

Becker, F., Montealecre, P., Rapaport, I., & Todinca, I. (2020). The Impact Of Locality In The Broadcast Congested Clique Model. SIAM Discret. Math., 34(1), 682–700.
Abstract: The broadcast congested clique model (BCLIQUE) is a messagepassing model of distributed computation where n nodes communicate with each other in synchronous rounds. First, in this paper we prove that there is a oneround, deterministic algorithm that reconstructs the input graph G if the graph is ddegenerate, and rejects otherwise, using bandwidth b = O(d . 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 BCLIQuE[r], we study the problem of detecting, in G, an induced cycle of length at most k (CYCLE <= k) and the problem of detecting an induced cycle of length at least k +1 (CYCLE>k). We give upper and lower bounds. We show that if each node is allowed to see up to distance r = left perpendicular k/2 right perpendicular + 1, then a polylogarithmic bandwidth is sufficient for solving CYCLE>k with only two rounds. Nevertheless, if nodes were allowed to see up to distance r = left perpendicular k/3 right perpendicular, then any oneround algorithm that solves CYCLE>k needs the bandwidth b to be at least Omega(n/ log n). We also show the existence of a oneround, deterministic BCLIQUE algorithm that solves CYCLE <= k with bandwitdh b = O(n(1/left perpendicular k/2 right perpendicular). log n). On the negative side, we prove that, if epsilon <= 1/3 and 0 < r <= k/4, then any epsilonerror, Rround, bbandwidth algorithm in the BCLIQUE[r] model that solves problem CYCLE(<= k )satisfies R . b = Omega(n(1/left perpendicular k/2 right perpendicular)).

Feuilloley, L., Fraigniaud, P., Montealegre, P., Rapaport, I., Remila, E., & Todinca, I. (2021). Compact Distributed Certification of Planar Graphs. Algorithmica, Early Access.
Abstract: Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACMSIAM symposium on discrete algorithms (SODA), pp 1096115, 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 lineartime 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 nnode networks. We show that a single interaction with the prover suffices, and randomization is unecessary, by providing an explicit description of a prooflabeling scheme for planarity, still using certificates on just O(log n) bits. We also show that there are no prooflabeling schemesin fact, even no locally checkable proofsfor planarity using certificates on o(log n) bits.

Fraigniaud, P., MontealegreBarba, P., Oshman, R., Rapaport, I., & Todinca, I. (2019). On Distributed MerlinArthur Decision Protocols. In Lecture Notes in Computer Sciences (Vol. 11639).

Goles, E., MontealegreBarba, P., & Todinca, I. (2013). The complexity of the bootstraping percolation and other problems. Theor. Comput. Sci., 504, 73–82.
Abstract: We study the problem of predicting the state of a vertex in automata networks, where the state at each site is given by the majority function over its neighborhood. We show that for networks with maximum degree greater than 5 the problem is PComplete, simulating a monotone Boolean circuit. Then, we show that the problem for networks with no vertex with degree greater than 4 is in NC, giving a fast parallel algorithm. Finally, we apply the result to the study of related problems. (C) 2012 Elsevier B.V. All rights reserved.

Liedloff, M., Montealegre, P., & Todinca, I. (2019). Beyond Classes of Graphs with “Few” Minimal Separators: FPT Results Through Potential Maximal Cliques. Algorithmica, 81(3), 986–1005.
Abstract: Let P(G,X) be a property associating a boolean value to each pair (G,X) where G is a graph and X is a vertex subset. Assume that P is expressible in counting monadic second order logic (CMSO) and let t be an integer constant. We consider the following optimization problem: given an input graph G=(V,E), find subsets XFV such that the treewidth of G[F] is at most t, property P(G[F],X) is true and X is of maximum size under these conditions. The problem generalizes many classical algorithmic questions, e.g., Longest Induced Path, Maximum Induced Forest, IndependentHPacking, etc. Fomin et al. (SIAM J Comput 44(1):5487, 2015) proved that the problem is polynomial on the class of graph Gpoly, i.e. the graphs having at most poly(n) minimal separators for some polynomial poly. Here we consider the class Gpoly+kv, formed by graphs of Gpoly to which we add a set of at most k vertices with arbitrary adjacencies, called modulator. We prove that the generic optimization problem is fixed parameter tractable on Gpoly+kv, with parameter k, if the modulator is also part of the input.

Montealegre, R., PerezSalazar, S., Rapaport, I., & Todinca, I. (2020). Graph reconstruction in the congested clique. J. Comput. Syst. Sci., 113, 1–17.
Abstract: In this paper we study the reconstruction problem in the congested clique model. Given a class of graphs g, the problem is defined as follows: if G is not an element of g, then every node must reject; if G is an element of g, then every node must end up knowing all the edges of G. The cost of an algorithm is the total number of bits received by any node through one link. It is not difficult to see that the cost of any algorithm that solves this problem is Omega(log vertical bar g(n)vertical bar/n), where g(n) is the subclass of all nnode labeled graphs in g. We prove that the lower bound is tight and that it is possible to achieve it with only 2 rounds. (C) 2020 Elsevier Inc. All rights reserved.

MontealegreBarba, P., PerezSalazar, S., Rapaport, I., & Todinca, I. (2018). Two Rounds Are Enough for Reconstructing Any Graph (Class) in the Congested Clique Model. In Lecture Notes in Computer Sciences (Vol. 11085).

Rapaport, I., Suchan, K., Todinca, I., & Verstraete, J. (2011). On Dissemination Thresholds in Regular and Irregular Graph Classes. Algorithmica, 59(1), 16–34.
Abstract: 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 < 1/2. We give families of graphs in which information spreads to all vertices with high probability for relatively small values of p.
