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D'Angelo, G., Di Stefano, G., Navarra, A., Nisse, N., & Suchan, K. (2015). Computing on Rings by Oblivious Robots: A Unified Approach for Different Tasks. Algorithmica, 72(4), 1055–1096.
Abstract: A set of autonomous robots have to collaborate in order to accomplish a common task in a ringtopology where neither nodes nor edges are labeled (that is, the ring is anonymous). We present a unified approach to solve three important problems: the exclusive perpetual exploration, the exclusive perpetual clearing, and the gathering problems. In the first problem, each robot aims at visiting each node infinitely often while avoiding that two robots occupy a same node (exclusivity property); in exclusive perpetual clearing (also known as graph searching), the team of robots aims at clearing the whole ring infinitely often (an edge is cleared if it is traversed by a robot or if both its endpoints are occupied); and in the gathering problem, all robots must eventually occupy the same node. We investigate these tasks in the LookComputeMove model where the robots cannot communicate but can perceive the positions of other robots. Each robot is equipped with visibility sensors and motion actuators, and it operates in asynchronous cycles. In each cycle, a robot takes a snapshot of the current global configuration (Look), then, based on the perceived configuration, takes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case it eventually moves to this neighbor (Move). Moreover, robots are endowed with very weak capabilities. Namely, they are anonymous, asynchronous, oblivious, uniform (execute the same algorithm) and have no common sense of orientation. In this setting, we devise algorithms that, starting from an exclusive and rigid (i.e. aperiodic and asymmetric) configuration, solve the three above problems in anonymous ringtopologies.

Feuilloley, L., Fraigniaud, P., Montealegre, P., Rapaport, I., Remila, E., & Todinca, I. (2021). Compact Distributed Certification of Planar Graphs. Algorithmica, 83(7), 2215–2244.
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., Montealegre, P., Rapaport, I., & Todinca, I. (2023). Alert Results A MetaTheorem for Distributed Certification 5 of 28 A MetaTheorem for Distributed Certification. Algorithmica, Early Access.
Abstract: Distributed certification, whether it be prooflabeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to each process in the system by a nontrustable oracle, and the processes are in charge of verifying these certificates, so that two properties are satisfied: completeness, i.e., for every legal instance, there is a certificate assignment leading all processes to accept, and soundness, i.e., for every illegal instance, and for every certificate assignment, at least one process rejects. The verification of the certificates must be fast, and the certificates themselves must be small. A large quantity of results have been produced in this framework, each aiming at designing a distributed certification mechanism for specific boolean predicates. This paper presents a “metatheorem”, applying to many boolean predicates at once. Specifically, we prove that, for every boolean predicate on graphs definable in the monadic secondorder (MSO) logic of graphs, there exists a distributed certification mechanism using certificates on O(log(2) n) bits in nnode graphs of bounded treewidth, with a verification protocol involving a single round of communication between neighbors

Kosowski, A., Li, B., Nisse, N., & Suchan, K. (2015). kChordal Graphs: From Cops and Robber to Compact Routing via Treewidth. Algorithmica, 72(3), 758–777.
Abstract: Cops and robber games, introduced by Winkler and Nowakowski (in Discrete Math. 43(23), 235239, 1983) and independently defined by Quilliot (in J. Comb. Theory, Ser. B 38(1), 8992, 1985), concern a team of cops that must capture a robber moving in a graph. We consider the class of kchordal graphs, i.e., graphs with no induced (chordless) cycle of length greater than k, ka parts per thousand yen3. We prove that k1 cops are always sufficient to capture a robber in kchordal graphs. This leads us to our main result, a new structural decomposition for a graph class including kchordal graphs. We present a polynomialtime algorithm that, given a graph G and ka parts per thousand yen3, either returns an induced cycle larger than k in G, or computes a treedecomposition of G, each bag of which contains a dominating path with at most k1 vertices. This allows us to prove that any kchordal graph with maximum degree Delta has treewidth at most (k1)(Delta1)+2, improving the O(Delta(Delta1) (k3)) bound of Bodlaender and Thilikos (Discrete Appl. Math. 79(13), 4561, 1997. Moreover, any graph admitting such a treedecomposition has small hyperbolicity). As an application, for any nvertex graph admitting such a treedecomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(klog Delta+logn) bits and achieving an additive stretch of O(klog Delta). As far as we know, this is the first routing scheme with O(klog Delta+logn)routing tables and small additive stretch for kchordal graphs.
Keywords: Treewidth; Chordality; Compact routing; Cops and robber games

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.
Keywords: FPT algorithms; Treewidth; Potential maximal cliques

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.
