Sequence submodular maximization meets streaming

• Published in: Journal of combinatorial optimization Vol. 41; no. 1; pp. 43 - 55
• Main Authors: Yang, Ruiqi, Xu, Dachuan, Guo, Longkun, Zhang, Dongmei
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2021; Springer Nature B.V
• Subjects: Combinatorics; Convex and Discrete Geometry; Graph theory; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Maximization; Operations Research/Decision Theory; Optimization; Sequences; Theory of Computation; Weight; Streaming algorithm; Sequence; Submodular maximization; Threshold
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-020-00662-5

In this paper, we study the problem of maximizing a sequence submodular function in the streaming setting, where the utility function is defined on sequences instead of sets of elements. We encode the sequence submodular maximization with a weighted digraph, in which the weight of a vertex reveals the utility value in selecting a single element and the weight of an edge reveals the additional profit with respect to a certain selection sequence. The edges are visited in a streaming fashion and the aim is to sieve a sequence of at most k elements from the stream, such that the utility is maximized. In this work, we present an edge-based threshold procedure, which makes one pass over the stream, attains an approximation ratio of ( 1 / ( 2 Δ + 1 ) - O ( ϵ ) ) , consumes O ( k Δ / ϵ ) memory source in total and O ( log ( k Δ ) / ϵ ) update time per edge, where Δ is the minimum of the maximal outdegree and indegree of the directed graph.