大型图中的随机行走控制：问题定义和快速求解
Random-walk domination in large graphs: problem definitions and fast solutions
摘要：
我们介绍并制定了两种类型的随机行走控制问题的图在实践中的一些应用（例如，在线社交网络中物品放置问题、广告网络中的广告放置问题以及P2P网络中的资源放置问题）。具体地，给定图$G$，第一类型的随机游走支配问题的目标是以$k$节点为目标，使得从剩余节点开始到目标节点的$L$长度随机游走的总命中时间最小。第二类随机游走支配问题是寻找$k$节点，以最大化通过$L$长度随机游走命中任何一个目标节点的预期节点数。证明了这两个问题是基数约束下的子模集函数极大化问题的两个特例。为了有效地解决这些问题，我们提出了一个基于动态规划（DP）的贪心算法，这是近最优的性能保证。然而，基于DP的贪心算法由于昂贵的边际增益评估而不是非常有效。为了进一步提高算法的速度，我们提出了一种近似的贪心算法，它具有线性的时间复杂度。图的大小，并且还具有接近最优的性能保证。近似贪心算法是基于一个精心设计的随机游走采样和样本物化技术。大量的实验表明，所提出的算法的有效性，效率和可扩展性。
英文摘要：
We introduce and formulate two types of random-walk domination problems in graphs motivated by a number of applications in practice (e.g., item-placement problem in online social network, Ads-placement problem in advertisement networks, and resource-placement problem in P2P networks). Specifically, given a graph $G$, the goal of the first type of random-walk domination problem is to target $k$ nodes such that the total hitting time of an $L$-length random walk starting from the remaining nodes to the targeted nodes is minimal. The second type of random-walk domination problem is to find $k$ nodes to maximize the expected number of nodes that hit any one targeted node through an $L$-length random walk. We prove that these problems are two special instances of the submodular set function maximization with cardinality constraint problem. To solve them effectively, we propose a dynamic-programming (DP) based greedy algorithm which is with near-optimal performance guarantee. The DP-based greedy algorithm, however, is not very efficient due to the expensive marginal gain evaluation. To further speed up the algorithm, we propose an approximate greedy algorithm with linear time complexity w.r.t.\ the graph size and also with near-optimal performance guarantee. The approximate greedy algorithm is based on a carefully designed random-walk sampling and sample-materialization techniques. Extensive experiments demonstrate the effectiveness, efficiency and scalability of the proposed algorithms.
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