DAG-Inducing Problems and Algorithms

In this paper, we show that in a parallel processing system, if a directed acyclic graph (DAG) can be induced in the state space and execution is \textit{enforced} along that DAG, then synchronization cost can be eliminated. Specifically, we show that in such systems, correctness is preserved even if the nodes execute asynchronously and rely on old/inconsistent information of other nodes. We present two variations for inducing DAGs -- \textit{DAG-inducing problems}, where the problem definition itself induces a DAG, and \textit{DAG-inducing algorithms}, where a DAG is induced by the algorithm. We demonstrate that the dominant clique (DC) problem and shortest path (SP) problem are DAG-inducing problems. Among these, DC allows self-stabilization, whereas the algorithm that we present for SP does not. We demonstrate that maximal matching (MM) and 2-approximation vertex cover (VC) are not DAG-inducing problems. However, DAG-inducing algorithms can be developed for them. Among these, the algorithm for MM allows self-stabilization and the 2-approx. algorithm for VC does not. Our algorithm for MM converges in $2n$ moves and does not require a synchronous environment, which is an improvement over the existing algorithms in the literature. Algorithms for DC, SP and 2-approx. VC converge in $2m$, $2m$ and $n$ moves respectively. We also note that DAG-inducing problems are more general than, and encapsulate, lattice linear problems (Garg, SPAA 2020). Similarly, DAG-inducing algorithms encapsulate lattice linear algorithms (Gupta and Kulkarni, SSS 2022).