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Alvarez-Miranda, E., & Pereira, J. (2019). On the complexity of assembly line balancing problems. Comput. Oper. Res., 108, 182–186.
Abstract: Assembly line balancing is a family of combinatorial optimization problems that has been widely studied in the literature due to its simplicity and industrial applicability. Most line balancing problems are NP-hard as they subsume the bin packing problem as a special case. Nevertheless, it is common in the line balancing literature to cite [A. Gutjahr and G. Nemhauser, An algorithm for the line balancing problem, Management Science 11 (1964) 308-315] in order to assess the computational complexity of the problem. Such an assessment is not correct since the work of Gutjahr and Nemhauser predates the concept of NP-hardness. This work points at over 50 publications since 1995 with the aforesaid error. (C) 2019 Elsevier Ltd. All rights reserved.
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Araujo, J., Ducoffe, G., Nisse, N., & Suchan, K. (2018). On interval number in cycle convexity. Discret. Math. Theor. Comput. Sci., 20(1), 35 pp.
Abstract: Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by in(cc)(G), is the minimum cardinality of a set S subset of V (G) such that every vertex w is an element of E V (G) \ S has two distinct neighbors u, v is an element of S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S. In this work, first we provide bounds on in(cc) (G) and its relations to other graph convexity parameters, and explore its behaviour on grids. Then, we present some hardness results by showing that deciding whetherin(cc) (G) <= k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtain that in(cc) (G) cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = NP). On the positive side, we present polynomial-time algorithms to compute in(cc) (G) for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q – 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G. Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.
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Fomin, F. V., Golovach, P. A., Kratochvil, J., Nisse, N., & Suchan, K. (2010). Pursuing a fast robber on a graph. Theor. Comput. Sci., 411(7-9), 1167–1181.
Abstract: The Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski and Winkler in the 1980s has been Much Studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves s time faster than the cops. We show that on split graphs, the problem is polynomially solvable if s = 1 but is NP-hard if s = 2. We further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for s <= 2. Finally, we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops. (C) 2009 Elsevier B.V. All rights reserved.
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Formenti, E., Goles, E., & Martin, B. (2012). Computational Complexity of Avalanches in the Kadanoff Sandpile Model. Fundam. Inform., 115(1), 107–124.
Abstract: This paper investigates the avalanche problem AP for the Kadanoff sandpile model (KSPM). We prove that (a slight restriction of) AP is in NC1 in dimension one, leaving the general case open. Moreover, we prove that AP is P-complete in dimension two. The proof of this latter result is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with an initial sand distribution in KSPM. These results are also related to the known prediction problem for sandpiles which is in NC1 for one-dimensional sandpiles and P-complete for dimension 3 or higher. The computational complexity of the prediction problem remains open for the Bak's model of two-dimensional sandpiles.
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Gajardo, A., & Goles, E. (2006). Crossing information in two-dimensional Sandpiles. Theor. Comput. Sci., 369(1-3), 463–469.
Abstract: We prove that in a two-dimensional Sandpile automaton, embedded in a regular infinite planar cellular space, it is impossible to cross information, if the bit of information is the presence (or absence) of an avalanche. This proves that it is impossible to embed arbitrary logical circuits in a Sandpile through quiescent configurations. Our result applies also for the non-planar neighborhood of Moore. Nevertheless, we also show that it is possible to compute logical circuits with a two-dimensional Sandpile, if a neighborhood of radius two is used in Z(2); crossing information becomes possible in that case, and we conclude that for this neighborhood the Sandpde is P-complete and Turing universal. (c) 2006 Elsevier B.V. All rights reserved.
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Gaspers, S., Liedloff, M., Stein, M., & Suchan, K. (2015). Complexity of splits reconstruction for low-degree trees. Discret Appl. Math., 180, 89–100.
Abstract: Given a vertex-weighted tree T, the split of an edge em T is the minimum over the weights of the two trees obtained by removing e from T, where the weight of a tree is the sum of weights of its vertices. Given a set of weighted vertices V and a multiset of integers s, we consider the problem of constructing a tree on V whose splits correspond to s. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be at most 4. We show that the problem is strongly NP-complete when T is required to be a path, the problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be at most 3, and it remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3. We also design polynomial time algorithms for the variant where T is required to be a path and the number of distinct vertex weights is constant, and the variant where all vertices have unit weight and T has a constant number of leaves. The latter algorithm is not only polynomial when the number of leaves, k, is a constant, but also is a fixed-parameter algorithm for parameter k. Finally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm. The considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong relation between structural invariants of the particles, such as the Wiener index and, less directly, the problem under consideration here, and physico-chemical properties of the substance. (C) 2014 Elsevier B.V. All rights reserved.
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Goles, E., & Montealegre, P. (2014). Computational complexity of threshold automata networks under different updating schemes. Theor. Comput. Sci., 559, 3–19.
Abstract: Given a threshold automata network, as well as an updating scheme over its vertices, we study the computational complexity associated with the prediction of the future state of a vertex. More precisely, we analyze two classes of local functions: the majority and the AND-OR rule (vertices take the AND or the OR logic functions over the state of its neighborhoods). Depending on the updating scheme, we determine the complexity class (NC, P, NP, PSPACE) where the prediction problem belongs. (C) 2014 Elsevier B.V. All rights reserved.
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Goles, E., & Montealegre, P. (2015). The complexity of the majority rule on planar graphs. Adv. Appl. Math., 64, 111–123.
Abstract: We study the complexity of the majority rule on planar automata networks. We reduce a special case of the Monotone Circuit Value Problem to the prediction problem of determining if a vertex of a planar graph will change its state when the network is updated with the majority rule. (C) 2014 Elsevier Inc. All rights reserved.
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Goles, E., & Montealegre, P. (2020). The complexity of the asynchronous prediction of the majority automata. Inf. Comput., 274(SI).
Abstract: We consider the asynchronous prediction problem for some automaton as the one consisting in determining, given an initial configuration, if there exists a non-zero probability that some selected site changes its state, when the network is updated picking one site at a time uniformly at random. We show that for the majority automaton, the asynchronous prediction problem is in NC in the two-dimensional lattice with von Neumann neighborhood. Later, we show that in three or more dimensions the problem is NP-Complete.
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Goles, E., & Moreira, A. (2012). Number-Conserving Cellular Automata and Communication Complexity: A Numerical Exploration Beyond Elementary CAs. J. Cell. Autom., 7(2), 151–165.
Abstract: We perform a numerical exploration of number-conserving cellular automata (NCCA) beyond the class of elementary CAs, in search of examples with high communication complexity. We consider some possible generalizations of the elementary rule 184 (a minimal model of traffic, which is the only non-trivial elementary NCCA). as well as the classes of NCCAs which minimally extend either the radius or the state set (with respect to the 2 states and radius 1 of the elementary case). Both for 3 states and radius 1, and for 2 stales and radius 2, NCCA appear that are conjectured to have maximal (exponential) communication complexity. Examples are given also for (conjectured) linear and quadratic behaviour.
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Goles, E., & Palacios, A. G. (2007). Dynamical complexity in cognitive neural networks. Biol. Res., 40(4), 479–485.
Abstract: In the last twenty years an important effort in brain sciences, especially in cognitive science, has been the development of mathematical tool that can deal with the complexity of extensive recordings corresponding to the neuronal activity obtained from hundreds of neurons. We discuss here along with some historical issues, advantages and limitations of Artificial Neural Networks (ANN) that can help to understand how simple brain circuits work and whether ANN can be helpful to understand brain neural complexity.
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Goles, E., Guillon, P., & Rapaport, I. (2011). Traced communication complexity of cellular automata. Theor. Comput. Sci., 412(30), 3906–3916.
Abstract: We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the state of the central cell will follow a given evolution, by communicating as little as possible between each other. We present some links with classical dynamical concepts, especially equicontinuity, expansivity, entropy and give the asymptotic communication complexity of most elementary cellular automata. (C) 2011 Elsevier B.V. All rights reserved.
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Goles, E., Lobos, F., Montealegre, P., Ruivo, E. L. P., & de Oliveira, P. P. B. (2020). Computational Complexity of the Stability Problem for Elementary Cellular Automata. J. Cell. Autom., 15(4), 261–304.
Abstract: Given an elementary cellular automaton and a cell v, we define the stability decision problem as the determination of whether or not the state of cell v will ever change, at least once, during the time evolution of the rule, over a finite input configuration. Here, we perform the study of the entire elementary cellular automata rule space, for the two possible decision cases of the problem, namely, changes in v from state 0 to 1 (0 -> 1), and the other way round (1 -> 0). Out of the 256 elementary cellular automata, we show that for all of them, at least one of the two decision problems is in the NC complexity class.
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Goles, E., Maldonado, D., Montealegre, P., & Ollinger, N. (2020). On the complexity of the stability problem of binary freezing totalistic cellular automata. Inf. Comput., 274, 21 pp.
Abstract: In this paper we study the family of two-state Totalistic Freezing Cellular Automata (TFCA) defined over the triangular and square grids with von Neumann neighborhoods. We say that a Cellular Automaton is Freezing and Totalistic if the active cells remain unchanged, and the new value of an inactive cell depends only on the sum of its active neighbors. We classify all the Cellular Automata in the class of TFCA, grouping them in five different classes: the Trivial rules, Turing Universal rules, Algebraic rules, Topological rules and Fractal Growing rules. At the same time, we study in this family the STABILITY problem, consisting in deciding whether an inactive cell becomes active, given an initial configuration. We exploit the properties of the automata in each group to show that: For Algebraic and Topological Rules the STABILITY problem is in NC. For Turing Universal rules the STABILITY problem is P-Complete. (C) 2020 Elsevier Inc. All rights reserved.
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Goles, E., Meunier, P. E., Rapaport, I., & Theyssier, G. (2011). Communication complexity and intrinsic universality in cellular automata. Theor. Comput. Sci., 412(1-2), 2–21.
Abstract: The notions of universality and completeness are central in the theories of computation and computational complexity. However, proving lower bounds and necessary conditions remains hard in most cases. In this article, we introduce necessary conditions for a cellular automaton to be “universal”, according to a precise notion of simulation, related both to the dynamics of cellular automata and to their computational power. This notion of simulation relies on simple operations of space-time rescaling and it is intrinsic to the model of cellular automata. intrinsic universality, the derived notion, is stronger than Turing universality, but more uniform, and easier to define and study. Our approach builds upon the notion of communication complexity, which was primarily designed to study parallel programs, and thus is, as we show in this article, particulary well suited to the study of cellular automata: it allowed us to show, by studying natural problems on the dynamics of cellular automata, that several classes of cellular automata, as well as many natural (elementary) examples, were not intrinsically universal. (C) 2010 Elsevier B.V. All rights reserved.
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Goles, E., Montealegre, P., Perrot, K., & Theyssier, G. (2018). On the complexity of two-dimensional signed majority cellular automata. J. Comput. Syst. Sci., 91, 1–32.
Abstract: We study the complexity of signed majority cellular automata on the planar grid. We show that, depending on their symmetry and uniformity, they can simulate different types of logical circuitry under different modes. We use this to establish new bounds on their overall complexity, concretely: the uniform asymmetric and the non-uniform symmetric rules are Turing universal and have a P-complete prediction problem; the non-uniform asymmetric rule is intrinsically universal; no symmetric rule can be intrinsically universal. We also show that the uniform asymmetric rules exhibit cycles of super-polynomial length, whereas symmetric ones are known to have bounded cycle length. (C) 2017 Elsevier Inc. All rights reserved.
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Goles, E., Moreira, A., & Rapaport, I. (2011). Communication complexity in number-conserving and monotone cellular automata. Theor. Comput. Sci., 412(29), 3616–3628.
Abstract: One third of the elementary cellular automata (CAs) are either number-conserving (NCCAs) or monotone (increasing or decreasing). In this paper we prove that, for all of them, we can find linear or constant communication protocols for the prediction problem. In other words, we are able to give a succinct description for their dynamics. This is not necessarily true for general NCCAs. In fact, we also show how to explicitly construct, from any CA, a new NCCA which preserves the original communication complexity. (C) 2011 Elsevier B.V. All rights reserved.
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Golovach, P. A., Heggernes, P., Lima, P. T., & Montealegre, P. (2020). Finding connected secluded subgraphs. J. Comput. Syst. Sci., 113, 101–124.
Abstract: 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 Pi-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 Pi, even when parameterized by k + t. We show that the situation changes when we are looking for a connected induced subgraph satisfying Pi. In particular, we show that the CONNECTED SECLUDED Pi-SUBGRAPH problem is FPT when parameterized by just t for many important graph properties Pi. (C) 2020 Elsevier Inc. All rights reserved.
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Mascareno, A., Goles, E., & Ruz, G. A. (2016). Crisis in complex social systems: A social theory view illustrated with the chilean case. Complexity, 21(S2), 13–23.
Abstract: The article argues that crises are a distinctive feature of complex social systems. A quest for connectivity of communication leads to increase systems' own robustness by constantly producing further connections. When some of these connections have been successful in recent operations, the system tends to reproduce the emergent pattern, thereby engaging in a non-reflexive, repetitive escalation of more of the same communication. This compulsive growth of systemic communication in crisis processes, or logic of excess, resembles the dynamic of self-organized criticality. Accordingly, we first construct the conceptual foundations of our approach. Second, we present three core assumptions related to the generative mechanism of social crises, their temporal transitions (incubation, contagion, restructuring), and the suitable modeling techniques to represent them. Third, we illustrate the conceptual approach with a percolation model of the crisis in Chilean education system. (c) 2016 Wiley Periodicals, Inc. Complexity 21: 13-23, 2016
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Montealegre, R., Perez-Salazar, 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 n-node 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.
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