Distributed Vertex Cover Reconfiguration

Abstract

Reconfiguration schedules, i.e., sequences that gradually transform one solution of a problem to another while always maintaining feasibility, have been extensively studied. Most research has dealt with the decision problem of whether a reconfiguration schedule exists, and the complexity of finding one. A prime example is the reconfiguration of vertex covers. We initiate the study of batched vertex cover reconfiguration, which allows to reconfigure multiple vertices concurrently while requiring that any adversarial reconfiguration order within a batch maintains feasibility. The latter provides robustness, e.g., if the simultaneous reconfiguration of a batch cannot be guaranteed. The quality of a schedule is measured by the number of batches until all nodes are reconfigured, and its cost, i.e., the maximum size of an intermediate vertex cover. To set a baseline for batch reconfiguration, we show that for graphs belonging to one of the classes $\{\mathsf{cycles, trees, forests, chordal, cactus, even\text{-}hole\text{-}free, claw\text{-}free}\}$, there are schedules that use $O(\varepsilon^{-1})$ batches and incur only a $1+\varepsilon$ multiplicative increase in cost over the best sequential schedules. Our main contribution is to compute such batch schedules in $O(\varepsilon^{-1}\log^* n)$ distributed time, which we also show to be tight. Further, we show that once we step out of these graph classes we face a very different situation. There are graph classes on which no efficient distributed algorithm can obtain the best (or almost best) existing schedule. Moreover, there are classes of bounded degree graphs which do not admit any reconfiguration schedules without incurring a large multiplicative increase in the cost at all.