Real-timesocial interactions andmulti-streamingare two critical features oflive streaming services. In this paper, we formulate a new fundamental service query,Social-aware Diverse and Preferred Organization Query (SDSQ), that jointly selects a set of diverse and preferred live streaming channels and a group of socially tight viewers for organization of a live multi-streaming soiree. We prove that SDSQ is NP-hard and inapproximable within any factor, and designSDSSel, a 2-approximation algorithm with a guaranteed error bound. Moreover, we study SDSQ-T, a special case of SDSQ, where the social graph is a threshold graph, and proposeTDSSel, a 2-approximation algorithm without any error to SDSQ-T. We propose two pruning strategies,PCPandCDPto boost SDSSel and TDSSel. We further propose a more challenging but practical service query,Generalized Social-aware Maximum Preferred and Diverse Query (GSPQ), a generalization of SDSQ. We designGPDSel, a 4-approximation algorithm for GSPQ with a guaranteed error bound. We propose a strategy to improve the approximation ratios of the proposed algorithms. A user study on Twitch validates SDSQ, and the large-scale experiments on real datasets demonstrate the superiority of the proposed algorithms over several baselines for live-streaming services.

Existing research on finding social groups mostly focuses on dense subgraphs in social networks. However, finding socially tenuous groups also has many important applications. In this paper, we introduce the notion of k-triangles to measure the tenuity of a group. We then formulate a new research problem, Minimum k-Triangle Disconnected Group with No-Pair Constraint (MkTG), to find a socially tenuous group from the online social network. We prove that MkTG is NP-hard and inapproximable within any ratio. Two algorithms, namely TERA and TERA-ADV, are designed for solving MkTG effectively and efficiently. Further, we examine the MkTG problem on tree-based social networks, due to their structural resemblance with corporate social networks built upon the supervision relation. Accordingly, we devise an efficient algorithm, namely Tenuity Maximization for Trees (TMT), to obtain the optimal solution in polynomial time. In addition, we study a more general version of MkTG, named Generalized Minimum k-Triangle Disconnected Group without No-Pair Constraint (MkTG-G). We formulate MkTG-G, analyze its inapproximability, and propose a randomized approximation algorithm, named Randomized Ranking with Limited Neighborhood Participation (RLNP). Experimental results on real datasets manifest that the proposed algorithms outperform the baselines in terms of both efficiency and solution quality.

Existing research on finding social groups mostly focuses on dense subgraphs in social networks. However, finding socially tenuous groups also has many important applications. In this paper, we introduce the notion of k-triangles to measure the tenuity of a group. We then formulate a new research problem, Minimum k-Triangle Disconnected Group (MkTG), to find a socially tenuous group from online social networks. We prove that MkTG is NP-Hard and inapproximable within any ratio in arbitrary graphs but polynomial-time tractable in threshold graphs. Two algorithms, namely TERA and TERA-ADV, are designed to exploit graph-theoretical approaches for solving MkTG on general graphs effectively and efficiently. Experimental results on seven real datasets manifest that the proposed algorithms outperform existing approaches in both efficiency and solution quality.

Although Software-Defined Networking (SDN) enables flexible network resource allocations for traffic engineering, current literature mostly focuses on unicast communications. Compared to traffic engineering for multiple unicast flows, multicast traffic engineering for multiple trees is very challenging not only because minimizing the bandwidth consumption of a single multicast tree by solving the Steiner tree problem is already NP-Hard, but the Steiner tree problem does not consider the link capacity constraint for multicast flows and node capacity constraint to store the forwarding entries in Group Table of OpenFlow. In this paper, therefore, we first study the hardness results of scalable multicast traffic engineering in SDN. We prove that scalable multicast traffic engineering with only the node capacity constraint is NP-Hard and not approximable within δ, which is the number of destinations in the largest multicast group. We then prove that scalable multicast traffic engineering with both the node and link capacity constraints is NP-Hard and not approximable within any ratio. To solve the problem, we design a δ-approximation algorithm, named Multi-Tree Routing and State Assignment Algorithm (MTRSA), for the first case and extend it to the general multicast traffic engineering problem. The simulation and implementation results demonstrate that the solutions obtained by the proposed algorithm outperform the shortest-path trees and Steiner trees. Most importantly, MTRSA is computation-efficient and can be deployed in SDN since it can generate the solution with numerous trees in a short time.

Research issues and data mining techniques for product recommendation and viral marketing have been widely studied. Existing works on seed selection in social networks do not take into account the effect of product recommendations in e-commerce stores. In this paper, we investigate the seed selection problem for viral marketing that considers both effects of social influence and item inference (for product recommendation). We develop a new model, Social Item Graph (SIG), that captures both effects in the form of hyperedges. Accordingly, we formulate a seed selection problem, called Social Item Maximization Problem (SIMP), and prove the hardness of SIMP. We design an efficient algorithm with performance guarantee, called Hyperedge-Aware Greedy (HAG), for SIMP and develop a new index structure, called SIG-index, to accelerate the computation of diffusion process in HAG. Moreover, to construct realistic SIG models for SIMP, we develop a statistical inference based framework to learn the weights of hyperedges from data. Finally, we perform a comprehensive evaluation on our proposals with various baselines. Experimental result validates our ideas and demonstrates the effectiveness and efficiency of the proposed model and algorithms over baselines.

The development and integration of social networking services and smartphones have made it easy for individuals to organize impromptu social activities anywhere and anytime. Main challenges arising in organizing impromptu activities are mostly due to the requirements of making timely invitations in accordance with the potential activity locations, corresponding to the locations of, and the relationships among the candidate attendees. Various combinations of candidate attendees and activity locations create a large solution space. Thus, in this paper, we propose Multiple Rally-Point Social Spatial Group Query (MRGQ), to select an appropriate activity location for a group of nearby attendees with tight social relationships. We first consider a special case of MRGQ, namely the Socio-Spatial Group Query (SSGQ), to determine a set of socially acquainted attendees while minimizing the total spatial distance to a specific activity location. We prove that SSGQ is NP-hard and formulate an Integer Linear Programming optimization model for SSGQ. We then develop an efficient algorithm, called SSGS, which employs effective pruning techniques to reduce the running time to determine the optimal solution. Moreover, we propose a heuristic algorithm for SSGQ to efficiently produce good solutions. We next consider the more general MRGQ. Although MRGQ is NP-hard, the number of attendees in practice is usually small enough such that an optimal solution can be found efficiently. Therefore, we first propose an Integer Linear Programming optimization model for MRGQ. We then design an efficient algorithm, called MAGS, which employs effective search space exploration and pruning strategies to reduce the running time for finding the optimal solution. We also propose to further optimize efficiency by indexing the potential activity locations. A user study demonstrates the strength of using SSGS and MAGS over manual coordination in terms of both solution quality and efficiency. Experimental results on real datasets show that our algorithms can process SSGQ and MRGQ efficiently and significantly outperform other baseline algorithms, including one based on the commercial parallel optimizer IBM CPLEX.

Current traffic engineering in SDN mostly focuses on unicast. By contrast, compared with individual unicast, multicast can effectively reduce network resources consumption to serve multiple clients jointly. Since many important applications require reliable transmissions, it is envisaged that reliable multicast plays a crucial role when an SDN operator plans to provide multicast services. However, the shortest-path tree (SPT) adopted in current Internet is not bandwidth-efficient, while the Steiner tree (ST) in Graph Theory is not designed to support reliable transmissions since the selection of recovery nodes is not examined. In this paper, therefore, we propose a new reliable multicast tree for SDN, named Recover-aware Steiner Tree (RST). The goal of RST is to minimize both tree and recovery costs, while finding an RST is very challenging. We prove that the RST problem is NP-Hard and inapproximable within k, which is the number of destination nodes. Thus, we design an approximate algorithm, called Recover Aware Edge Reduction Algorithm (RAERA), to solve the problem. The simulation results on real networks and large synthetic networks, together with the experiment on our SDN testbed with real YouTube traffic, all manifest that RST outperforms both SPT and ST. Also, the implementation of RAERA in SDN controllers shows that an RST can be returned within a few seconds and thereby is practical for SDN networks.

Software-Defined Networking (SDN) enables flexible network resource allocations for traffic engineering, but at the same time the scalability problem becomes more serious since traffic is more difficult to be aggregated. Those crucial issues in SDN have been studied for unicast but have not been explored for multicast traffic, and addressing those issues for multicast is more challenging since the identities and the number of members in a multicast group can be arbitrary. In this paper, therefore, we propose a new multicast tree for SDN, named Branch-aware Steiner Tree (BST). The BST problem is difficult since it needs to jointly minimize the numbers of the edges and the branch nodes in a tree, and we prove that it is NP-Hard and inapproximable within k, which denotes the number of group members. We further design an approximation algorithm, called Branch Aware Edge Reduction Algorithm (BAERA), to solve the problem. Simulation results demonstrate that the trees obtained by BAERA are more bandwidth-efficient and scalable than the shortest-path trees and traditional Steiner trees. Most importantly, BAERA is computation-efficient to be deployed in SDN since it can generate a tree on massive networks in small time.

Let $G$ be a graph with a threshold function $\theta:V(G)\rightarrow \mathbb{N}$ such that $1\leq \theta(v)\leq d_G(v)$ for every vertex $v$ of $G$, where $d_G(v)$ is the degree of $v$ in $G$. Suppose we are given a target set $S\subseteq V(G)$. This paper considers the following repetitive process on $G$. At time step $0$ the vertices of $S$ are colored black and the other vertices are colored white. After that, at each time step $t>0$, the colors of white vertices (if any) are updated according to the following rule. All white vertices $v$ that have at least $\theta(v)$ black neighbors at the time step $t-1$ are colored black, and the colors of the other vertices do not change. The process runs until no more white vertices can update colors from white to black. The following optimization problem is called Target Set Selection: Find a target set $S$ of smallest possible size such that all vertices in $G$ are black at the end of the process. Such an $S$ is called an optimal target set for $G$ under the threshold function $\theta$. We are interested in finding an optimal target set for the well-known class of honeycomb networks under an important threshold function called a strict majority threshold, where $\theta(v)=\lceil (d_G(v)+1)/2\rceil$ for each vertex $v$ in $G$. In a graph $G$, a feedback vertex set is a subset $S\subseteq V(G)$ such that the subgraph induced by $V(G)\setminus S$ is cycle-free. In this paper we give exact value on the size of the optimal target set for various kinds of honeycomb networks under a strict majority threshold, and as a by-product we also provide a minimum feedback vertex set for different kinds of regular graphs in the class of honeycomb networks.