We propose a new concept named adaptive submodularity ratio to study the greedy policy for sequential decision making. While the greedy policy is known to perform well for a wide variety of adaptive stochastic optimization problems in practice, its theoretical properties have been analyzed only for a limited class of problems. We narrow the gap between theory and practice by using adaptive submodularity ratio, which enables us to prove approximation guarantees of the greedy policy for a substantially wider class of problems. Examples of newly analyzed problems include important applications such as adaptive influence maximization and adaptive feature selection. Our adaptive submodularity ratio also provides bounds of adaptivity gaps. Experiments confirm that the greedy policy performs well with the applications being considered compared to standard heuristics.
Influence maximization was proposed by . An adaptive version of influence maximization was first considered by . They showed that this objective function satisfies adaptive submodularity under the independent cascade model in general graphs. Influence maximization on a bipartite graph has been studied for applications to advertisement selection @cite_18 @cite_15 . This problem setting was extended to the adaptive setting by , but only the independent cascade model was considered. The curvature of its objective function was studied by .
@cite_18 @cite_15
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We consider the budget allocation problem over bipartite influence model proposed by (, 2012). This problem can be viewed as the well-known influence maximization problem with budget constraints. We first show that this problem and its much more general form fall into a general setting; namely the monotone submodular function maximization over integer lattice subject to a knapsack constraint. Our framework includes 's model, even with a competitor and with cost. We then give a (1 - 1 e)-approximation algorithm for this more general problem. Furthermore, when influence probabilities are nonincreasing, we obtain a faster (1 - 1 e)-approximation algorithm, which runs essentially in linear time in the number of nodes. This allows us to implement our algorithm up to almost 10M edges (indeed, our experiments tell us that we can implement our algorithm up to 1 billion edges. It would approximately take us only 500 seconds.). Influence maximization is a hard combinatorial optimization problem. It requires the identification of an optimum set of k network vertices that triggers the activation of a maximum total number of remaining network nodes with respect to a chosen propagation model. The problem is appealing because it is provably hard and has a number of practical applications in domains such as data mining and social network analysis. Although there are many exact and heuristic algorithms for influence maximization, it has been tackled by metaheuristic and evolutionary methods as well. This paper presents and evaluates a new evolutionary method for influence maximization that employs a recent genetic algorithm for fixed–length subset selection. The algorithm is extended by the concept of guiding that prevents selection of infeasible vertices, reduces the search space, and effectively improves the evolutionary procedure. In this paper, we consider the problem of maximizing the spread of influence through a social network. Given a graph with a threshold value thr(v) attached to each vertex v, the spread of influence is modeled as follows: A vertex v becomes ''active'' (influenced) if at least thr(v) of its neighbors are active. In the corresponding optimization problem the objective is then to find a fixed number k of vertices to activate such that the number of activated vertices at the end of the propagation process is maximum. We show that this problem is strongly inapproximable in time f(k)@?n^O^(^1^), for some function f, even for very restrictive thresholds. In the case that the threshold of each vertex equals its degree, we prove that the problem is inapproximable in polynomial time and it becomes r(n)-approximable in time f(k)@?n^O^(^1^), for some function f, for any strictly increasing function r. Moreover, we show that the decision version parameterized by k is W[1]-hard but becomes fixed-parameter tractable on bounded degree graphs. In the well-studied Influence Maximization problem, the goal is to identify a set of k nodes in a social network whose joint influence on the network is maximized. A large body of recent work has justified research on Influence Maximization models and algorithms with their potential to create societ al or economic value. However, in order to live up to this potential, the algorithms must be robust to large amounts of noise, for they require quantitative estimates of the influence, which individuals exert on each other; ground truth for such quantities is inaccessible, and even decent estimates are very difficult to obtain. We begin to address this concern formally. First, we exhibit simple inputs on which even very small estimation errors may mislead every algorithm into highly suboptimal solutions. Motivated by this observation, we propose the Perturbation Interval model as a framework to characterize the stability of Influence Maximization against noise in the inferred diffusion network. Analyzing the susceptibility of specific instances to estimation errors leads to a clean algorithmic question, which we term the Influence Difference Maximization problem. However, the objective function of Influence Difference Maximization is NP-hard to approximate within a factor of O(n1−e) for any e > 0. Given the infeasibility of diagnosing instability algorithmically, we focus on finding influential users robustly across multiple diffusion settings. We define a Robust Influence Maximization framework wherein an algorithm is presented with a set of influence functions. The algorithm’s goal is to identify a set of k nodes who are simultaneously influential for all influence functions, compared to the (function-specific) optimum solutions. We show strong approximation hardness results for this problem unless the algorithm gets to select at least a logarithmic factor more seeds than the optimum solution. However, when enough extra seeds may be selected, we show that techniques of can be used to approximate the optimum robust influence to within a factor of 1−1 e. We evaluate this bicriteria approximation algorithm against natural heuristics on several real-world datasets. Our experiments indicate that the worst-case hardness does not necessarily translate into bad performance on real-world datasets; all algorithms perform fairly well.