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Bitar, N., Goles, E., & Montealegre, P. (2022). COMPUTATIONAL COMPLEXITY OF BIASED DIFFUSION-LIMITED AGGREGATION. SIAM Discret. Math., 36(1), 823–866.
Abstract: Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote by k-DLA the model where the particles move only in k possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid c and a sequence S of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site c when sequence S is realized. The second problem is Realization, where the input is a set of positions of the grid, P. The question is whether there exists a sequence S that realizes P, i.e. all particles of S exactly occupy the positions in P. Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is P-Complete for 2-DLA (thus for 3-DLA). Later, we show that Prediction can be solved much more efficiently for 1-DLA. In fact, we show that in that case the problem is NL-Complete. With respect to Realization, we show that restricted to 2-DLA the problem is in P, while in the 1-DLA case, the problem is in L.
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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., 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. (2021). On the complexity of asynchronous freezing cellular automata. Inf. Comput., 281, 104764.
Abstract: In this paper we study the family of freezing cellular automata (FCA) in the context of asynchronous updating schemes. A cellular automaton is called freezing if there exists an order of its states, and the transitions are only allowed to go from a lower to a higher state. A cellular automaton is asynchronous if at each time-step only one cell is updated. We define the problem ASYNCUNSTABILITY, which consists in deciding there exists a sequential updating scheme that changes the state of a given cell.
We begin showing that ASYNCUNSTABILITY is in NL for any one-dimensional FCA. Then we focus on the family of life-like freezing CA (LFCA). We study the complexity of ASYNCUNSTABILITY for all LFCA in the triangular and square grids, showing that almost all of them can be solved in NC, except for one rule for which the problem is NP-Complete. (C) 2021 Elsevier Inc. All rights reserved.
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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., Montalva-Medel, M., Montealegre, P., & Rios-Wilson, M. (2022). On the complexity of generalized Q2R automaton. Adv. Appl. Math., 138, 102355.
Abstract: We study the dynamic and complexity of the generalized Q2R automaton. We show the existence of non-polynomial cycles as well as its capability to simulate with the synchronous update the classical version of the automaton updated under a block sequential update scheme. Furthermore, we show that the decision problem consisting in determine if a given node in the network changes its state is P-Hard.
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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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