Sequence Submodular Maximization Meets Streaming

Abstract

In this paper, we consider the streaming sequence submodular maximization problem, in which the utility function is defined on sequences of element instead of element sets. We encode the values of different sequences by a weighted directed acyclic graph (W-DAG), where the weight of vertex reveals the utility value in selecting single element and the weight of an edge proclaims the additional benefit according to a certain selected order. In addition, the edges are revealed in a streaming fashion that one edge is known in one time slot. The aim is to output a sequence of vertices of length bounded by a given constant k, such that the utility function value is maximized. In this work, we first provide the framework of the sequence submodular maximization under streaming. By utilizing an edge-based thresholding principle, we derive a one pass, $(1-2\varDelta /(2\varDelta +1- \varepsilon ))$-approximation algorithm with $O(\varepsilon ^{-1}k\varDelta \log (k\varDelta ))$ memory and $O(\varepsilon ^{-1}\log (k\varDelta ))$ update time per edge, where $\varDelta =\min \{\varDelta _{\text {in}},\varDelta _{\text {out}}\}$ and $\varDelta _{\text {in}}$ ($\varDelta _{\text {out}}$) is the maximum in-degree (out-degree) of the constructed W-DAG. At last, we present a further improved streaming algorithm, which also requires one pass over the stream and attains the same approximation ratio but is with a decreased memory complexity of $O(\varepsilon ^{-1}k\varDelta )$.