arXiv subjects

Pablo Concha-Vega

Publications and source records attributed to Pablo Concha-Vega.

5 recordsLinked to original sources

arXiv2026

Non-Uniform and Weighted Crossing Gates in Two-Dimensional Sandpiles

Determining whether predicting two-dimensional sandpiles lies in $\mathbf{NC}$ or is $\mathbf{P}$-complete has been open for decades. Moore and Nilsson proved $\mathbf{P}$-completeness for the three dimensional case by encoding Boolean circuits into sandpiles, but this method fails in two dimension due to the impossibility of crossing gates. In this work, we study the existence of crossing gates on non-uniform and weighted grids. We establish an equivalence between uniform weighted crossing gates and a class of simple non-uniform crossing gates, which we call primal. We also exhibit a crossing gate that inherently requires more than one crossing, rather than a single crossing as in standard constructions. Finally, we show that the equivalence between uniform weighted and primal crossings breaks down in more general settings.

cs.CC↗

arXiv2026

Local Homophily on Bicolored Graphs is $\mathbf{P}$-complete

We propose a local transformation on bicolored graphs, which we call local homophily, inspired by adaptive networks and based on majority dynamics and homophily. In this transformation, a vertex updates its color to match the majority of its neighbors, while neighbors of the same color become connected and neighbors of the opposite color become disconnected. We show how to simulate Boolean circuits using local homophily and establish that determining whether a given pair of vertices becomes connected under iterative applications of local homophily is $\mathbf{P}$-complete under logspace reductions.

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arXiv2025

Complexity of the Freezing Majority Rule with L-shaped Neighborhoods

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 $\mathsf{NC}$, 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 $\mathsf{P}$-complete, indicating that the problem might be inherently sequential.

cs.CC↗

arXiv2025

Timed Prediction Problem for Sandpile Models

We investigate the computational complexity of the timed prediction problem in two-dimensional sandpile models. This question refines the classical prediction problem, which asks whether a cell q will eventually become unstable after adding a grain at cell p from a given configuration. The prediction problem has been shown to be P-complete in several settings, including for subsets of the Moore neighborhood, but its complexity for the von Neumann neighborhood remains open. In a previous work, we provided a complete characterization of crossover gates (a key to the implementation of non-planar monotone circuits) for these small neighborhoods, leading to P-completeness proofs with only 4 and 5 neighbors among the eight adjancent cells. In this paper, we introduce the timed setting, where the goal is to determine whether cell q becomes unstable exactly at time t. We distinguish several cases: some neighborhoods support complete timed toolkits (including timed crossover gates) and exhibit P-completeness; others admit timed crossovers but suffer from synchronization issues; planar neighborhoods provably do not admit any timed crossover; and finally, for some remaining neighborhoods, we conjecture that no timed crossover is possible.

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arXiv2025

Is Graph Local Complementation Inherently Sequential?

Local complementation of a graph $G$ on vertex $v$ is an operation that results in a new graph $G*v$, where the neighborhood of $v$ is complemented. Two graph are locally equivalent if on can be reached from the other one through local complementation. It was previously established that recognizing locally equivalent graphs can be done in $\mathcal{O}(n^4)$ time. We sharpen this result by proving it can be decided in $\mathcal{O}(\log^2(n))$ parallel time with $n^{\mathcal{O}(1)}$ processors. As a second contribution, we introduce the Local Complementation Problem, a decision problem that captures the complexity of applying a sequence of local complementations. Given a graph $G$, a sequence of vertices $s$, and a pair of vertices $u,v$, the problem asks whether the edge $(u,v)$ is present in the graph obtained after applying local complementations according to $s$. Regardless it simplicity, it is proven to be $\mathsf{P}$-complete, therefore it is unlikely to be efficiently parallelizable. Finally, it is conjectured that Local Complementation Problem remains $\mathsf{P}$-complete when restricted to circle graphs.

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