parallel algorithm designers

Abstract: We prove that prediction for the synchronous two-dimensional majority rule is $\mathrm{P}$-complete under logspace many-one reductions, resolving a problem open for almost three decades. In 1997, Moore established $\mathrm{P}$-completeness in dimension three and higher and conjectured that the two-dimensional case admits an efficient parallel algorithm. We consider an $n\times n$ torus on which each cell follows the majority of its four nearest neighbors and retains its current state in a tie. Given an explicitly specified initial configuration and a time $T$, prediction asks whether a designated cell is in state $+1$ at time $T$. The central challenge is to make independent information streams cross in the plane under a homogeneous, monotone, diffusive local rule. We overcome this obstacle through a temporal encoding of Boolean values: both values generate activity, but are distinguished by signal arrival times. This encoding yields a crossover that preserves both values and composes with wires, duplication, and AND and OR gates to simulate arbitrary monotone Boolean circuits. Thus a local rule that favors agreement can nevertheless transport, combine, and cross independent information in two dimensions. We also prove $\mathrm{P}$-completeness for deciding whether a designated cell ever reaches $+1$, without a prescribed time horizon. The prediction result extends to every uniform symmetric signed majority rule on the same neighborhood, including the minority rule.

Abstract: The $k$-core of a graph is its (unique) largest subgraph with minimum degree at least $k$. For any $k \geq 3$, deciding whether a given vertex belongs to the $k$-core is a P-complete problem, meaning that it is inherently sequential and highly unlikely to admit efficient parallel algorithms, even on graphs of maximum degree $k+1$. This paper investigates alternative parameterizations of the $k$-core problem to identify conditions under which it can be placed into sub-polynomial complexity classes. We prove that the problem is in para-NC$^{2+ε}$ when parameterized by treewidth, and in para-NC$^3$ when parameterized by $k$ on chordal graphs. Furthermore, we introduce a novel NC$^{3}$ algorithm for interval graphs when $k = \mathcal{O}(\lg v(G))$, which relies on an improved parameterization by pathwidth. Finally, we establish corresponding lower bounds, demonstrating that, even with these parameterizations, computing the $k$-core remains L-hard, meaning it requires at least logarithmic space. These findings explore the boundary of parallel tractability for the $k$-core problem by highlighting the graph parameters that make it inherently sequential.