The problem of influence maximization is to select the most influential individuals in a social network. With the popularity of social network sites, and the development of viral marketing, the importance of the problem has been increased. The influence maximization problem is NP-hard, and therefore, there will not exist a polynomial-time algorithm to solve the problem unless P=NP. Many heuristics are proposed to find a nearly good solution in a shorter time. In this paper, we propose two heuristic algorithms to find good solutions. The heuristics are based on two ideas: (1) vertices of high degree have more influence in the network, and (2) nearby vertices influence on almost analogous sets of vertices. We evaluate our algorithms on several well-known data sets and show that our heuristics achieve better results (up to @math in influence spread) for this problem in a shorter time (up to @math improvement in the running time).
Chen al @cite_8 proposed new greedy algorithms for independent cascade and weighted cascade models. They made the greedy algorithm faster by combining their algorithms with CELF. They also proposed a new heuristic, named , which produces results of quality close to the greedy algorithm, while is much faster than that and performs better than the traditional degree and distance centrality heuristics. In order to avoid running repeated influence propagation simulations, Borgs al @cite_16 generated a random hypergraph according to reverse reachability probability of vertices in the original graph and select @math vertices that cover the largest number of vertices in the hypergraph. They guarantee @math approximation ratio of the solution with probability at least @math . Later, Tang al @cite_14 @cite_15 proposed TIM and IMM to cover drawbacks of Borgs al's algorithm @cite_16 and improved its running time. Bucur and Iacca @cite_1 and Kr "o mer and Nowakov 'a @cite_11 used genetic algorithms for the influence maximization problem. Weskida and Michalski @cite_6 used GPU acceleration in their genetic algorithm to improve its efficiency.
@cite_14 @cite_8 @cite_1 @cite_6 @cite_15 @cite_16 @cite_11
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Nowadays, in the world of limited attention, the techniques that maximize the spread of social influence are more than welcomed. Companies try to maximize their profits on sales by providing customers with free samples believing in the power of word-of-mouth marketing, governments and non-governmental organizations often want to introduce positive changes in the society by appropriately selecting individuals or election candidates want to spend least budget yet still win the election. In this work we propose the use of evolutionary algorithm as a mean for selecting seeds in social networks. By framing the problem as genetic algorithm challenge we show that it is possible to outperform well-known greedy algorithm in the problem of influence maximization for the linear threshold model in both: quality (up to 16 better) and efficiency (up to 35 times faster). We implemented these two algorithms by using GPGPU approach showing that also the evolutionary algorithm can benefit from GPU acceleration making it efficient and scaling better than the greedy algorithm. As the experiments conducted by using three real world datasets reveal, the evolutionary approach proposed in this paper outperforms the greedy algorithm in terms of the outcome and it also scales much better than the greedy algorithm when the network size is increasing. The only drawback in the GPGPU approach so far is the maximum size of the network that can be processed - it is limited by the memory of the GPU card. We believe that by showing the superiority of the evolutionary approach over the greedy algorithm, we will motivate the scientific community to look for an idea to overcome this limitation of the GPU approach - we also suggest one of the possible paths to explore. Since the proposed approach is based only on topological features of the network, not on the attributes of nodes, the applications of it are broader than the ones that are dataset-specific. The greedy sequential algorithm for maximal independent set (MIS) loops over the vertices in an arbitrary order adding a vertex to the resulting set if and only if no previous neighboring vertex has been added. In this loop, as in many sequential loops, each iterate will only depend on a subset of the previous iterates (i.e. knowing that any one of a vertex's previous neighbors is in the MIS, or knowing that it has no previous neighbors, is sufficient to decide its fate one way or the other). This leads to a dependence structure among the iterates. If this structure is shallow then running the iterates in parallel while respecting the dependencies can lead to an efficient parallel implementation mimicking the sequential algorithm. In this paper, we show that for any graph, and for a random ordering of the vertices, the dependence length of the sequential greedy MIS algorithm is polylogarithmic (O(log^2 n) with high probability). Our results extend previous results that show polylogarithmic bounds only for random graphs. We show similar results for greedy maximal matching (MM). For both problems we describe simple linear-work parallel algorithms based on the approach. The algorithms allow for a smooth tradeoff between more parallelism and reduced work, but always return the same result as the sequential greedy algorithms. We present experimental results that demonstrate efficiency and the tradeoff between work and parallelism. Influence maximization, defined as finding a small subset of nodes that maximizes spread of influence in social networks, is NP-hard under both Linear Threshold (LT) and Independent Cascade (IC) models, where a line of greedy heuristic algorithms have been proposed. The simple greedy algorithm [14] achieves an approximation ratio of 1-1 e. The advanced CELF algorithm [16], by exploiting the sub modular property of the spread function, runs 700 times faster than the simple greedy algorithm on average. However, CELF is still inefficient [4], as the first iteration calls for N times of spread estimations (N is the number of nodes in networks), which is computationally expensive especially for large networks. To this end, in this paper we derive an upper bound function for the spread function. The bound can be used to reduce the number of Monte-Carlo simulation calls in greedy algorithms, especially in the first iteration of initialization. Based on the upper bound, we propose an efficient Upper Bound based Lazy Forward algorithm (UBLF in short), by incorporating the bound into the CELF algorithm. We test and compare our algorithm with prior algorithms on real-world data sets. Experimental results demonstrate that UBLF, compared with CELF, reduces more than 95 Monte-Carlo simulations and achieves at least 2-5 times speed-raising when the seed set is small. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger.