Abstract
Social Influence Maximization Problems (SIMPs) deal with selecting k seeds in a given Online Social Network (OSN) to maximize the number of eventually-influenced users. This is done by using these seeds based on a given set of influence probabilities among neighbors in the OSN. Although the SIMP has been proved to be NP-hard, it has both submodular (with a natural diminishing-return) and monotone (with an increasing influenced users through propagation) that make the problem suitable for approximation solutions. However, several special SIMPs cannot be modeled as submodular or monotone functions. In this paper, we look at several conditions under which non-submodular or non-monotone functions can be handled or approximated. One is a profit-maximization SIMP where seed selection cost is included in the overall utility function, breaking the monotone property. The other is a crowd-influence SIMP where crowd influence exists in addition to individual influence, breaking the submodular property. We then review several new techniques and notions, including double-greedy algorithms and the supermodular degree, that can be used to address special SIMPs. Our main results show that for a specific SIMP model, special network structures of OSNs can help reduce its time complexity of the SIMP.
Highlights
This section reviews the notion of the submodular function with associated properties
Many optimization problems in combinatorics, graphs, and game theory can be represented as non-negative submodular functions
An optimization problem concerning a convex or concave function can be described as a problem of maximizing or minimizing a submodular function with or without constraints
Summary
Many optimization problems in combinatorics, graphs, and game theory can be represented as non-negative submodular functions. Tsinghua Science and Technology, December 2020, 25(6): 703–711 number of uncovered elements) This greedy cover is a ln n C 1 and 1 1=e approximation of the set cover and maximum coverage problems, respectively. Let us look at two cases: Submodular but non-monotone: The max-cut problem is as follows: in the unweighted version, we are given an undirected graph, and our goal is to partition the graph into two node sets to maximize the number of edges crossing these two sets. Monotone but non-submodular: The welfare maximization problem is as follows: there is a set of players and a set of indivisible items. Each player has his own (monotone, non-decreasing) valuation for any subset of items. A simple greedy algorithm that iteratively maximizes the marginal gain, obtains an approximation ratio of 1 1=e to the optimal algorithm
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