On the approximability of positive influence dominating set in social networks

In social networks, there is a tendency for connected users to match each other’s behaviors. Moreover, a user likely adopts a behavior, if a certain fraction of his family and friends follows that behavior. Identifying people who have the most influential effect to the others is of great advantages, especially in politics, marketing, behavior correction, and so on. Under a graph-theoretical framework, we study the positive influence dominating set (PIDS) problem that seeks for a minimal set of nodes $\mathcal{P}$ such that all other nodes in the network have at least a fraction ρ>0 of their neighbors in $\mathcal{P}$ . We also study a different formulation, called total positive influence dominating set (TPIDS), in which even nodes in $\mathcal{P}$ are required to have a fraction ρ of neighbors inside $\mathcal{P}$ . We show that neither of these problems can be approximated within a factor of (1??)lnmax{Δ,|V|^1/2}, where Δ is the maximum degree. Moreover, we provide a simple proof that both problems can be approximated within a factor lnΔ+O(1). In power-law networks, where the degree sequence follows a power-law distribution, both problems admit constant factor approximation algorithms. Finally, we present a linear-time exact algorithms for trees.     

An improved distributed data aggregation scheduling in wireless sensor networks

This paper focuses on the distributed data aggregation collision-free scheduling problem, which is one of very important issues in wireless sensor networks. Bo et al. (Proc. IEEE INFOCOM, 2009) proposed an approximate distributed algorithm for the problem and Xu et al. (Proc. ACM FOWANC, 2009) proposed a centralized algorithm and its distributed implementation to generate a collision-free scheduling for the problem, which are the only two existing distributed algorithms. Unfortunately, there are a few mistakes in their performance analysis in Bo et al. (Proc. IEEE INFOCOM, 2009), and the distributed algorithm can not get the same latency as the centralized algorithm because the distributed implementation was not an accurate implementation of the centralized algorithm (Xu et al. in Proc. ACM FOWANC, 2009). According to those, we propose an improved distributed algorithm to generate a collision-free schedule for data aggregation in wireless sensor networks. Not an arbitrary tree in Bo et al. (Proc. IEEE INFOCOM, 2009) but a breadth first search tree (BFS) rooted at the sink node is adopted, the bounded latency 61R+5Δ?67 of the schedule is obtained, where R is the radius of the network with respect to the sink node and Δ is the maximum node degree. We also correct the latency bound of the schedule in Bo et al. (Proc. IEEE INFOCOM, 2009) as 61D+5Δ?67, where D is a diameter of the network and prove that our algorithm is more efficient than the algorithm (Bo et al. in Proc. IEEE INFOCOM, 2009). We also give a latency bound for the distributed implementation in Xu et al. (Proc. ACM FOWANC, 2009).     

The extremal spectral radii of $$$$-uniform supertrees

In this paper, we study some extremal problems of three kinds of spectral radii of \(k\)-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence \(Q\)-spectral radius). We call a connected and acyclic \(k\)-uniform hypergraph a supertree. We introduce the operation of “moving edges” for hypergraphs, together with the two special cases of this operation: the edge-releasing operation and the total grafting operation. By studying the perturbation of these kinds of spectral radii of hypergraphs under these operations, we prove that for all these three kinds of spectral radii, the hyperstar \(\mathcal {S}_{n,k}\) attains uniquely the maximum spectral radius among all \(k\)-uniform supertrees on \(n\) vertices. We also determine the unique \(k\)-uniform supertree on \(n\) vertices with the second largest spectral radius (for these three kinds of spectral radii). We also prove that for all these three kinds of spectral radii, the loose path \(\mathcal {P}_{n,k}\) attains uniquely the minimum spectral radius among all \(k\)-th power hypertrees of \(n\) vertices. Some bounds on the incidence \(Q\)-spectral radius are given. The relation between the incidence \(Q\)-spectral radius and the spectral radius of the matrix product of the incidence matrix and its transpose is discussed.     

Minimum diameter cost-constrained Steiner trees

Given an edge-weighted undirected graph $G=(V,E,c,w)$ where each edge $e\in E$ has a cost $c(e)\ge 0$ and another weight $w(e)\ge 0$ , a set $S\subseteq V$ of terminals and a given constant $\mathrm{C}_0\ge 0$ , the aim is to find a minimum diameter Steiner tree whose all terminals appear as leaves and the cost of tree is bounded by $\mathrm{C}_0$ . The diameter of a tree refers to the maximum weight of the path connecting two different leaves in the tree. This problem is called the minimum diameter cost-constrained Steiner tree problem, which is NP-hard even when the topology of the Steiner tree is fixed. In this paper, we deal with the fixed-topology restricted version. We prove the restricted version to be polynomially solvable when the topology is not part of the input and propose a weakly fully polynomial time approximation scheme (weakly FPTAS) when the topology is part of the input, which can find a $(1+\epsilon )$ –approximation of the restricted version problem for any $\epsilon >0$ with a specific characteristic.     

Distance-d independent set problems for bipartite and chordal graphs

The paper studies a generalization of the Independent Set problem (IS for short). A distance- $d$ independent set for an integer $d\ge 2$ in an unweighted graph $G = (V, E)$ is a subset $S\subseteq V$ of vertices such that for any pair of vertices $u, v \in S$ , the distance between $u$ and $v$ is at least $d$ in $G$ . Given an unweighted graph $G$ and a positive integer $k$ , the Distance- $d$ Independent Set problem (D $d$ IS for short) is to decide whether $G$ contains a distance- $d$ independent set $S$ such that $|S| \ge k$ . D2IS is identical to the original IS. Thus D2IS is $\mathcal{NP}$ -complete even for planar graphs, but it is in $\mathcal{P}$ for bipartite graphs and chordal graphs. In this paper we investigate the computational complexity of D $d$ IS, its maximization version MaxD $d$ IS, and its parameterized version ParaD $d$ IS( $k$ ), where the parameter is the size of the distance- $d$ independent set: (1) We first prove that for any $\varepsilon >0$ and any fixed integer $d\ge 3$ , it is $\mathcal{NP}$ -hard to approximate MaxD $d$ IS to within a factor of $n^{1/2-\varepsilon }$ for bipartite graphs of $n$ vertices, and for any fixed integer $d\ge 3$ , ParaD $d$ IS( $k$ ) is $\mathcal{W}[1]$ -hard for bipartite graphs. Then, (2) we prove that for every fixed integer $d\ge 3$ , D $d$ IS remains $\mathcal{NP}$ -complete even for planar bipartite graphs of maximum degree three. Furthermore, (3) we show that if the input graph is restricted to chordal graphs, then D $d$ IS can be solved in polynomial time for any even $d\ge 2$ , whereas D $d$ IS is $\mathcal{NP}$ -complete for any odd $d\ge 3$ . Also, we show the hardness of approximation of MaxD $d$ IS and the $\mathcal{W}[1]$ -hardness of ParaD $d$ IS( $k$ ) on chordal graphs for any odd $d\ge 3$ .     

Minimum number of disjoint linear forests covering a planar graph

Graph coloring has interesting real-life applications in optimization, computer science and network design, such as file transferring in a computer network, computation of Hessians matrix and so on. In this paper, we consider one important coloring, linear arboricity, which is an improper edge coloring. Moreover, we study linear arboricity on planar graphs with maximum degree \(\varDelta \ge 7\) . We have proved that the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \) , if for each vertex \(v\in V(G)\) , there are two integers \(i_v,j_v\in \{3,4,5,6,7,8\}\) such that any two cycles of length \(i_v\) and \(j_v\) , which contain \(v\) , are not adjacent. Clearly, if \(i_v=i, j_v=j\) for each vertex \(v\in V(G)\) , then we can easily get one corollary: for two fixed integers \(i,j\in \{3,4,5,6,7,8\}\) , if there is no adjacent cycles with length \(i\) and \(j\) in \(G\) , then the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \) .     

Robustness of power-law networks: its assessment and optimization

Many practical complex networks, such as the Internet, WWW and social networks, are discovered to follow power-law distribution in their degree sequences, i.e., the number of nodes with degree \(i\) in these networks is proportional to \(i^{-\beta }\) for some exponential factor \(\beta > 0\). However, these networks also expose their vulnerabilities to a great number of threats such as adversarial attacks on the Internet, cyber-crimes on the WWW or malware propagations on social networks. Although power-law networks have been found robust under random attacks and vulnerable to intentional attacks via experimental observations, how to better understand their vulnerabilities from a theoretical point of view still remains open. In this paper, we study the vulnerability of power-law networks under random attacks and adversarial attacks using the in-depth probabilistic analysis on the theory of random power-law graph models. Our results indicate that power-law networks are able to tolerate random failures if their exponential factor \(\beta \) is \(<\)2.9, and they are more robust against intentional attacks if \(\beta \) is smaller. Furthermore, we reveal the best range \([1.8, 2.5]\) for the exponential factor \(\beta \) by optimizing the complex networks in terms of both their vulnerabilities and costs. When \(\beta < 1.8\), the network maintenance cost is very expensive, and when \(\beta > 2.5\) the network robustness is unpredictable since it depends on the specific attacking strategy.     

Radiation hybrid map construction problem parameterized

In this paper, we study the Radiation hybrid map construction ( $\mathsf{{RHMC} }$ ) problem which is about reconstructing a genome from a set of gene clusters. The problem is known to be $\mathsf{{NP} }$ -complete even when all gene clusters are of size two and the corresponding problem ( $\mathsf{{RHMC}_2 }$ ) admits efficient constant-factor approximation algorithms. In this paper, for the first time, we consider the more general case when the gene clusters can have size either two or three ( $\mathsf{{RHMC}_3 }$ ). Let ${p\text{- }\mathsf {RHMC} }$ be a parameterized version of $\mathsf{{RHMC} }$ where the parameter is the size of solution. We present a linear kernel for ${p\text{- }\mathsf {RHMC}_3 }$ of size $22k$ that when combined with a bounded search-tree algorithm, gives an FPT algorithm running in $O(6^kk+n)$ time. For ${p\text{- }\mathsf {RHMC}_3 }$ we present a bounded search tree algorithm which runs in $O^*(2.45^k)$ time, greatly improving the previous bound using weak kernels.     

Online scheduling with rejection and reordering: exact algorithms for unit size jobs

We study an online scheduling problem with rejection on \(m\ge 2\) identical machines, in which we deal with unit size jobs. Each arriving job has a rejection value (a rejection cost or penalty for minimization problems, and a rejection profit for maximization problems) associated with it. A buffer of size \(K\) is available to store \(K\) jobs. A job which is not stored in the buffer must be either assigned to a machine or rejected. Upon the arrival of a new job, the job can be stored in the buffer if there is a free slot (possibly created by evicting other jobs and assigning or rejecting every evicted job). At termination, the buffer must be emptied. We study four variants of the problem, as follows. We study the makespan minimization problem, where the goal is to minimize the sum of the makespan and the penalty of rejected jobs, and the \(\ell _p\) norm minimization problem, where the goal is to minimize the sum of the \(\ell _p\) norm of the vector of machine completion times and the penalty of rejected jobs. We also study two maximization problems, where the goal in the first version is to maximize the sum of the minimum machine load (the cover value of the machines) and the total rejection profit, and in the second version the goal is to maximize a function of the machine completion times (which measures the balance of machine loads) and the total rejection profit. We show that an optimal solution (an exact solution for the offline problem) can always be obtained in this environment, and determine the required buffer size. Specifically, for all four variants we present optimal algorithms with \(K=m-1\) and prove that in each case, using a buffer of size at most \(m-2\) does not allow the design of an optimal algorithm, which makes our algorithms optimal in this respect as well. The lower bounds hold even for the special case where the rejection value is equal for all input jobs.     

Algorithms for the minimum diameter terminal Steiner tree problem

Given an undirected connected graph \(G=(V(G),E(G),d)\) with a function \(d(\cdot )\ge 0\) on edges and a subset \(S\subseteq V(G)\) of terminals, the minimum diameter terminal Steiner tree problem (MDTSTP) asks for a terminal Steiner tree in \(G\) of a minimum diameter. In the paper, the diameter of a tree refers to the longest of all the distances between two different leaves of the tree. When \(G\) is a complete graph and \(d(\cdot )\) is a metric function, we demonstrate that an optimal solution of MDTSTP is monopolar or dipolar and give an \(O(|S|\cdot |V(G)\setminus S|^2)\) -time exact algorithm. For the nonmetric version of MDTSTP, we present a simple 2-approximation algorithm with a time complexity of \(O(|V(G)\setminus S|\log |S|)\) , as well as two exact algorithms with a time complexity of \(O(|S|^3|V(G)|^2)\) and \(O(|S|\cdot |V(G)\setminus S|^2+|S|^2\cdot |V(G)\setminus S|)\) , respectively.     

Online maximum directed cut

We investigate a natural online version of the well-known Maximum Directed Cut problem on DAGs. We propose a deterministic algorithm and show that it achieves a competitive ratio of $\frac{3\sqrt{3}}{2}\approx 2.5981$ . We then give a lower bound argument to show that no deterministic algorithm can achieve a ratio of $\frac{3\sqrt{3}}{2}-\epsilon$ for any ??>0 thus showing that our algorithm is essentially optimal. Then, we extend our technique to improve upon the analysis of an old result: we show that greedily derandomizing the trivial randomized algorithm for MaxDiCut in general graphs improves the competitive ratio from 4 to 3, and also provide a tight example.     

Efficient algorithms for the max k -vertex cover problem

Given a graph  \(G(V,E)\) of order  \(n\) and a constant \(k \leqslant n\) , the max  \(k\) -vertex cover problem consists of determining  \(k\) vertices that cover the maximum number of edges in  \(G\) . In its (standard) parameterized version, max  \(k\) -vertex cover can be stated as follows: “given  \(G,\) \(k\) and parameter  \(\ell ,\) does  \(G\) contain  \(k\) vertices that cover at least  \(\ell \) edges?”. We first devise moderately exponential exact algorithms for max  \(k\) -vertex cover, with time-complexity exponential in  \(n\) but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for max  \(k\) -vertex cover with complexity bounded above by the maximum among  \(c^k\) and  \(\gamma ^{\tau },\) for some \(\gamma < 2,\) where  \(\tau \) is the cardinality of a minimum vertex cover of  \(G\) (note that \({\textsc {max}}\,\) k \({\textsc {\!-vertex cover}}{} \notin \mathbf{FPT}\) with respect to parameter  \(k\) unless \(\mathbf{FPT} = \mathbf{W[1]}\) ), using polynomial space. We finally study approximation of max  \(k\) -vertex cover by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catch-up on polynomial inapproximability, by providing algorithms achieving, with worst-case running times importantly smaller than those needed for exact computation, approximation ratios unachievable in polynomial time.     

Computing an effective decision making group of a society using social network analysis

Recent years have witnessed how much a decision making group can be dysfunctional due to the extreme hyperpartisanship. While partisanship is crucial for the representatives to pursue the wishes of those whom they represent for, such an extremism results in a severe gridlock in the decision making progress, and makes themselves highly inefficient. It is known that such a problem can be mitigated by having negotiators in the group. This paper investigates the potential of social network analysis techniques to choose an effective leadership group of a society such that it suffers less from the extreme hyperpartisanship. We establish three essential requirements for an effective representative group, namely Influenceability, Partisanship, and Bipartisanship. Then, we formulate the problem of finding a minimum size representative group satisfying the three requirements as the minimum connected \(k\) -core dominating set problem (MC \(k\) CDSP), and show its NP-hardness. We introduce an extension of MC \(k\) CDSP, namely MC \(k\) CDSP-C, which assumes the society has a number of sub-communities and requires at least one representative from each sub-community should be in the leadership. We also propose an approximation algorithm for a subclass of MC \(k\) CDSP with \(k=2\) , and show an \(\alpha \) -approximation algorithm of MC \(k\) CDSP can be used to obtain an \(\alpha \) -approximation algorithm of MC \(k\) CDSP-SC.     

A greedy algorithm for the fault-tolerant connected dominating set in a general graph

Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless network is an effective way to save energy and alleviate broadcasting storm. Since nodes may fail due to an accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. A node set \(C\) is an \(m\) -fold connected dominating set ( \(m\) -fold CDS) of graph \(G\) if every node in \(V(G)\setminus C\) has at least \(m\) neighbors in \(C\) and the subgraph of \(G\) induced by \(C\) is connected. In this paper, we will present a greedy algorithm to compute an \(m\) -fold CDS in a general graph, which has size at most \(2+\ln (\Delta +m-2)\) times that of a minimum \(m\) -fold CDS, where \(\Delta \) is the maximum degree of the graph. This result improves on the previous best known performance ratio of \(2H(\Delta +m-1)\) for this problem, where \(H(\cdot )\) is the Harmonic number.     

On certain geometric properties of the Yao–Yao graphs

We show that, for any constant $\rho > 1$ , there exists an integer constant $k$ such that the Yao–Yao graph with parameter $k$ defined on a civilized unit disk graph is a geometric spanner of stretch factor $\rho $ . This improves the results of Wang and Li in several aspects, as described in the paper. This partially answers an open problem posed by Demaine, Mitchell and O’Rourke about the spanner properties of Yao–Yao graphs. We also show that the Yao–Yao graph with parameter $k=4$ defined on the complete Euclidean graph is not plane.     

Mobile facility location: combinatorial filtering via weighted occupancy

An instance of the mobile facility location problem consists of a complete directed graph \(G = (V, E)\) , in which each arc \((u, v) \in E\) is associated with a numerical attribute \(\mathcal M (u,v)\) , representing the cost of moving any object from \(u\) to \(v\) . An additional ingredient of the input is a collection of servers \(S = \{ s_1, \ldots , s_k \}\) and a set of clients \(C = \{ c_1, \ldots , c_\ell \}\) , which are located at nodes of the underlying graph. With this setting in mind, a movement scheme is a function \(\psi : S \rightarrow V\) that relocates each server \(s_i\) to a new position, \(\psi ( s_i )\) . We refer to \(\mathcal M ( s_i, \psi ( s_i ) )\) as the relocation cost of \(s_i\) , and to \(\min _{i \in [k]} \mathcal M (c_j, \psi ( s_i ) )\) , the cost of assigning client \(c_j\) to the nearest final server location, as the service cost of \(c_j\) . The objective is to compute a movement scheme that minimizes the sum of relocation and service costs. In this paper, we resolve an open question posed by Demaine et al. (SODA ’07) by characterizing the approximability of mobile facility location through LP-based methods. We also develop a more efficient algorithm, which is based on a combinatorial filtering approach. The latter technique is of independent interest, as it may be applicable in other settings as well. In this context, we introduce a weighted version of the occupancy problem, for which we establish interesting tail bounds, not before demonstrating that existing bounds cannot be extended.     

Minimum-latency aggregation scheduling in wireless sensor network

Data aggregation is an essential yet time-consuming task in wireless sensor networks. This paper studies the well-known minimum-latency aggregation schedule problem and proposes an energy-efficient distributed scheduling algorithm named Clu-DDAS based on a novel cluster-based aggregation tree. Our approach differs from all the previous schemes where connected dominating sets or maximal independent sets are employed. This paper proves that Clu-DDAS has a latency bound of \(4R'+2 \varDelta -2\), where \(\varDelta \) is the maximum degree and \(R'\) is the inferior network radius which is smaller than the network radius \(R\). Our experiments show that Clu-DDAS has an approximate latency upper bound of \(4R'+1.085 \varDelta -2\) with increased \(\varDelta \). Clu-DDAS has comparable latency as the previously best centralized algorithm, E-PAS, but consumes 78 % less energy as shown by the simulation results. Clu-DDAS outperforms the previously best distributed algorithm, DAS, whose latency bound is \(16R'+\varDelta -14\) on both latency and energy consumption. On average, Clu-DDAS transmits 67 % fewer total messages than DAS. The paper also proposes an adaptive strategy for updating the schedule to accommodate dynamic network topology.     

Approximating soft-capacitated facility location problem with uncertainty

In this paper we devise the stochastic and robust approaches to study the soft-capacitated facility location problem with uncertainty. We first present a new stochastic soft-capacitated model called The 2-Stage Soft Capacitated Facility Location Problem and solve it via an approximation algorithm by reducing it to linear-cost version of 2-stage facility location problem and dynamic facility location problem. We then present a novel robust model of soft-capacitated facility location, The Robust Soft Capacitated Facility Location Problem. To solve it, we improve the approximation algorithm proposed by Byrka et al. (LP-rounding algorithms for facility-location problems. CoRR, 2010a) for RFTFL and then treat it similarly as in the stochastic case. The improvement results in an approximation factor of \(\alpha + 4\) for the robust fault-tolerant facility location problem, which is best so far.     

The number of edges,spectral radius and Hamilton-connectedness of graphs

In this paper, we prove that a simple graph G of order sufficiently large n with the minimal degree \(\delta (G)\ge k\ge 2\) is Hamilton-connected except for two classes of graphs if the number of edges in G is at least \(\frac{1}{2}(n^2-(2k-1)n + 2k-2)\). In addition, this result is used to present sufficient spectral conditions for a graph with large minimum degree to be Hamilton-connected in terms of spectral radius or signless Laplacian spectral radius, which extends the results of (Zhou and Wang in Linear Multilinear Algebra 65(2):224–234, 2017) for sufficiently large n. Moreover, we also give a sufficient spectral condition for a graph with large minimum degree to be Hamilton-connected in terms of spectral radius of its complement graph.     

F_{3}-domination problem of graphs

Given a graph \(G\) and a set \(S\subseteq V(G),\) a vertex \(v\) is said to be \(F_{3}\) -dominated by a vertex \(w\) in \(S\) if either \(v=w,\) or \(v\notin S\) and there exists a vertex \(u\) in \(V(G)-S\) such that \(P:wuv\) is a path in \(G\) . A set \(S\subseteq V(G)\) is an \(F_{3}\) -dominating set of \(G\) if every vertex \(v\) is \(F_{3}\) -dominated by a vertex \(w\) in \(S.\) The \(F_{3}\) -domination number of \(G\) , denoted by \(\gamma _{F_{3}}(G)\) , is the minimum cardinality of an \(F_{3}\) -dominating set of \(G\) . In this paper, we study the \(F_{3}\) -domination of Cartesian product of graphs, and give formulas to compute the \(F_{3}\) -domination number of \(P_{m}\times P_{n}\) and \(P_{m}\times C_{n}\) for special \(m,n.\)