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This paper deals with the complexity of some natural graph problems parameterized by some measures that are restrictions of clique-width, such as modular-width and neighborhood diversity. We introduce a novel parameter, called iterated type partition number, that can be computed in linear time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W [ 1 ] -hard when parameterized by the iterated type partition number. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parameterized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity. Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition number, which enables us to find optimal solutions for several graph problems. As an example, in this paper, we present algorithms parameterized by the iterated type partition number of the input graph for some generalized versions of the Maximum Clique , Minimum Graph Coloring , (Total) Minimum Dominating Set , Minimum Vertex Cover and Maximum Independent Set problems. Each algorithm outputs not only the optimal value but also the optimal solution. We stress that while the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. We finally show that the proposed scheme can be used to devise polynomial kernels, with respect to iterated type partition number, for the decisional version of most of the problems mentioned above. [ABSTRACT FROM AUTHOR]

We consider the problem of keeping under control the spread of harmful items in networks, such as the contagion proliferation of diseases or the diffusion of fake news. We assume the linear threshold model of diffusion where each node has a threshold that measures the node's resistance to the contagion. We study the parameterized complexity of the problem: Given a network, a set of initially contaminated nodes, and two integers k and ℓ , is it possible to limit the diffusion to at most k other nodes of the network by immunizing at most ℓ nodes? We consider several parameters associated with the input, including the bounds k and ℓ , the maximum node degree Δ , the number ζ of initially contaminated nodes, the treewidth, and the neighborhood diversity of the network. We first give W[1] or W[2]-hardness results for each of the considered parameters. Then we give fixed-parameter algorithms for some parameter combinations. [ABSTRACT FROM AUTHOR]

We introduce a novel domination problem, which we call Dual Domination (DD). We assume that the nodes in a network are partitioned into two categories: Positive nodes (V +) and negative nodes (V −). We study the Maximum Bounded Dual Domination , where, given a bound k , the problem is to find a set D ⊆ V + , which maximizes the number of nodes dominated in V + , dominating at most k nodes in V −. We show that such problem is hard to approximate to a factor better than (1 − 1 / e). We give a polynomial time algorithm with approximation guaranteed (1 − e − 1 / Δ) , where Δ represents the maximum number of neighbors in V + of any node in V − , and an O (| V | k 2) time algorithm to solve the problem on trees. We also study two related problems named Maximum Dual Domination and minimum Negative Dual Domination. For both problems, we show hardness results and provide O (| V |) time algorithms on trees. [ABSTRACT FROM AUTHOR]

A widely studied model of influence diffusion in social networks represents the network as a graph G = (V , E) , with an integer influence threshold t (v) for each node, and the diffusion process as follows: Initially the members of a chosen set S ⊆ V are influenced and, during each subsequent round, the set of influenced nodes is augmented by including every new node v that has at least t (v) previously influenced neighbours. The general problem is to find a small initial set that influences the whole network. In this paper we extend this model by using incentives to reduce the thresholds of some nodes. The goal is to minimize the total amount of the incentive required to ensure that the information diffusion process terminates within a given number of rounds λ. The problem is hard to approximate in general networks. We present optimal polynomial-time algorithms for paths, cycles, trees, and complete networks for any λ. For the special case λ = 1 , we present a polynomial-time algorithm with a logarithmic approximation guarantee for any network. [ABSTRACT FROM AUTHOR]

Alice wants to join a new social network, and influence its members to adopt a new product or idea. Each person v in the network has a certain threshold t (v) for activation , i.e. adoption of the product or idea. If v has at least t (v) activated neighbors, then v will also become activated. If Alice wants to activate the entire social network, whom should she befriend? More generally, we study the problem of finding the minimum number of links that a set of external influencers should form to people in the network, in order to activate the entire social network. This Minimum Links Problem has applications in viral marketing and the study of epidemics. Its solution can be quite different from the related and widely studied Target Set Selection problem. We prove that the Minimum Links problem cannot be approximated to within a ratio of O (2 log 1 − ϵ ⁡ n) , for any fixed ϵ > 0 , unless N P ⊆ D T I M E (n p o l y l o g (n)) , where n is the number of nodes in the network. On the positive side, we give linear time algorithms to solve the problem for trees, cycles, and cliques, for any given set of external influencers, and give precise bounds on the number of links needed. For general graphs, we design a polynomial time algorithm to compute size-efficient link sets that can activate the entire graph. [ABSTRACT FROM AUTHOR]

A (k , n) -superimposed code is a well known and widely used combinatorial structure that can be represented by a t × n binary matrix such that for any k columns of the matrix and for any column c chosen among these k columns, there exists a row in correspondence of which column c has an entry equal to 1 and the remaining k − 1 columns have entries equal to 0. Due to the many situations in which superimposed codes find applications, there is an abundant literature that studies the problem of constructing (k , n) -superimposed codes with a small number t of rows. Motivated by applications to conflict-free communication in multiple-access networks, group testing, and data security, we study the problem of constructing superimposed codes that have the additional constraints that the number of 1's in each column of the matrix is constant, and equal to an input parameter w. Our results improve on the known literature in the area. We also extend our findings to other important combinatorial structures, like selectors, generalized superimposed codes, and z -error correcting superimposed codes. [ABSTRACT FROM AUTHOR]

Given two discrete random variables $X$ and $Y$ , with probability distributions $p=(p_{1}, \ldots , p_{n})$ and $q=(q_{1}, \ldots , q_{m})$ , respectively, let us denote by ${\mathcal{ C}}(p, q)$ the set of all couplings of $p$ and $q$ , that is, the set of all bivariate probability distributions that have $p$ and $q$ as marginals. In this paper, we study the problem of finding a joint probability distribution in ${\mathcal{ C}}(p, q)$ of minimum entropy (equivalently, a coupling that maximizes the mutual information between $X$ and $Y$), and we discuss several situations where the need for this kind of optimization naturally arises. Since the optimization problem is known to be NP-hard, we give an efficient algorithm to find a joint probability distribution in ${\mathcal{ C}}(p, q)$ with entropy exceeding the minimum possible at most by 1 bit, thus providing an approximation algorithm with an additive gap of at most 1 bit. Leveraging on this algorithm, we extend our result to the problem of finding a minimum-entropy joint distribution of arbitrary $k\geq {2}$ discrete random variables $X_{1}, \ldots , X_{k}$ , consistent with the known $k$ marginal distributions of the individual random variables $X_{1}, \ldots , X_{k}$. In this case, our algorithm has an additive gap of at most $\log k$ from optimum. We also discuss several related applications of our findings and extensions of our results to entropies different from the Shannon entropy. [ABSTRACT FROM AUTHOR]

Abstract Identifying the most influential spreaders is an important issue for the study of the dynamics of information diffusion in complex networks. In this paper we analyze the following spreading model. Initially, a few nodes know a piece of information and are active spreaders of it. At subsequent rounds, spreaders communicate the information to their neighbors. Upon receiving the information, a node becomes aware of it but does not necessarily become a spreader; it starts spreading only if it gets the information from a sufficiently large number of its neighbors. A set of initial spreaders that guarantees that all the nodes become aware of the information is called a perfect seed set. We study the problem of choosing a perfect seed set of minimum size. We provide hardness results and show that the problem becomes tractable on trees. In case of general graphs, we provide an efficient algorithm and validate its effectiveness (in terms of the solution size) on real-life networks. We also study perfect seed sets in dense graphs and derive a bound on the size of a dominating set in Ore graphs. [ABSTRACT FROM AUTHOR]

Given a network represented by a graph G=(V,E), we consider a dynamical process of influence diffusion in G that evolves as follows: Initially only the nodes of a given S⊆V are influenced; subsequently, at each round, the set of influenced nodes is augmented by all the nodes in the network that have a sufficiently large number of already influenced neighbors. The question is to determine a small subset of nodes S (a target set) that can influence the whole network. This is a widely studied problem that abstracts many phenomena in the social, economic, biological, and physical sciences. It is known that the above optimization problem is hard to approximate within a factor of 2log1-ϵ|V|, for any ϵ>0. In this paper, we present a fast and surprisingly simple algorithm that exhibits the following features: (1) when applied to trees, cycles, or complete graphs, it always produces an optimal solution (i.e, a minimum size target set); (2) when applied to arbitrary networks, it always produces a solution of cardinality which improves on previously known upper bounds; (3) when applied to real-life networks, it always produces solutions that substantially outperform the ones obtained by previously published algorithms (for which no proof of optimality or performance guarantee is known in any class of graphs). [ABSTRACT FROM AUTHOR]

It is well known that the entropy $H(X)$ of a discrete random variable $X$ is always greater than or equal to the entropy $H(f(X))$ of a function $f$ of $X$ , with equality if and only if $f$ is one-to-one. In this paper, we give tight bounds on $H(f(X))$ , when the function $f$ is not one-to-one, and we illustrate a few scenarios, where this matters. As an intermediate step toward our main result, we derive a lower bound on the entropy of a probability distribution, when only a bound on the ratio between the maximal and minimal probabilities is known. The lower bound improves on previous results in the literature, and it could find applications outside the present scenario. [ABSTRACT FROM AUTHOR]

We study a combinatorial model of the spread of influence in networks that generalizes existing schemata recently proposed in the literature. In our model, agents change behaviours/opinions on the basis of information collected from their neighbours in a time interval of bounded size whereas agents are assumed to have unbounded memory in previously studied scenarios. In our mathematical framework, one is given a network G = ( V , E ) , an integer value t ( v ) for each node v ∈ V , and a time window size λ . The goal is to determine a small set of nodes ( target set ) that influences the whole graph. The spread of influence proceeds in rounds as follows: initially all nodes in the target set are influenced; subsequently, in each round, any uninfluenced node v becomes influenced if the number of its neighbours that have been influenced in the previous λ rounds is greater than or equal to t ( v ) . We prove that the problem of finding a minimum cardinality target set that influences the whole network G is hard to approximate within a polylogarithmic factor. On the positive side, we design exact polynomial time algorithms for paths, rings, and trees. [ABSTRACT FROM AUTHOR]

We consider conflict-free colorings of graph neighborhoods: Each vertex of the graph must be assigned a color so that for each vertex v there is at least one color appearing exactly once in the neighborhood of v . The goal is to minimize the number of used colors. We consider both the case of closed neighborhoods, when the neighborhood of a node includes the node itself, and the case of open neighborhoods when a node does not belong to its neighborhood. In this paper, we study complexity aspects of the problem. We show that the problem of conflict-free coloring of closed neighborhoods is NP-complete. Moreover, we give non-approximability results for the conflict-free coloring of open neighborhoods. From a positive point of view, both problems become tractable if parameterized by the vertex cover number or the neighborhood diversity number of the graph. We present simple algorithms which improve on existing results. [ABSTRACT FROM AUTHOR]

We consider the $$k$$ -strong conflict-free ( $$k$$ -SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval $$I$$ of the family there are at least $$k$$ colors each appearing exactly once in $$I$$ . We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when $$k=1$$ and $$5-\frac{2}{k}$$ when $$k\ge 2$$ . In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any $$k \ge 1$$ . We also provide, in case $$k=O({{\mathrm{polylog}}}(n))$$ , a quasipolynomial time algorithm to decide the existence of a $$k$$ -SCF coloring that uses at most $$q$$ colors. [ABSTRACT FROM AUTHOR]

Abstract: A wireless network consists of a large number of devices, deployed over a geographical area, and of a base station where data sensed by the devices are collected and accessed by the end users. In this paper we study algorithmic and complexity issues originating from the problem of data gathering in wireless networks. We give an algorithm to construct minimum makespan transmission schedules for data gathering under the following hypotheses: the communication graph is a tree network, the transmissions in the network can interfere with each other up to distance , where , and no buffering is allowed at intermediate nodes. In the interesting case in which all nodes in the network have to deliver an arbitrary non-zero number of packets, we provide a closed formula for the makespan of the optimal gathering schedule. Additionally, we consider the problem of determining the computational complexity of data gathering in general graphs and show that the problem is NP-complete. On the positive side, we design a simple -factor approximation algorithm for general networks. [Copyright &y& Elsevier]

The article presents information on a research paper on discrete mathematics which focuses on two problems in combinatorial optimization namely spider cover problem and maximum spider problem. It refers to two goals in addressing these problems namely to find a spider that covers most elements in the family and minimum number of spiders covering all subsets in the family. It makes use of approximation algorithms with best possible approximation ratios in solving the problems.

Data gathering is a fundamental operation in wireless sensor networks in which data packets generated at sensor nodes are to be collected at a base station. In this paper we suppose that each sensor is equipped with an half-duplex interface; hence, a node cannot receive and transmit at the same time. Moreover, each node is equipped with omnidirectional antennas allowing the transmission over distance R. The network is a multi-hop wireless network and the time is slotted so that one-hop transmission of one data item consumes one time slot. We model the network with a graph where the vertices represent the nodes and two nodes are connected if they are in the transmission range of each other. We suppose that the interference range is the same as the transmission range; therefore due to interferences a collision happens at a node if two or more of its neighbors try to transmit at the same time. Furthermore we suppose that an intermediate node should forward a message as soon as it receives it. We give an optimal collision free gathering schedule for tree networks whenever each node has exactly. [ABSTRACT FROM AUTHOR]

Abstract: We study Small-World graphs in the perspective of their use in the development of efficient as well as easy to implement network infrastructures. Our analysis starts from the Small-World model proposed by Kleinberg: a grid network augmented with directed long-range random links. The choices of the long-range links are independent from one node to another. In this setting greedy routing and some of its variants have been analyzed and shown to produce paths of polylogarithmic expected length. We start from asking whether all the randomness, used in Kleinberg’s model for establishing the long-range contacts of the nodes, is indeed necessary to assure the existence of short paths. In order to deal with the above question, we impose (stringent) restrictions on the choice of long-range links and we show that such restrictions do not increase the average path length of greedy routing and its variations. We are able to decrease the number of random bits, required to establish each node’s long-range link, from to on a network of size . Diminishing the randomness in the choice of random links has several benefits; in particular, it implies an increase in the clustering of the graph, thus increasing the resilience of the network. [Copyright &y& Elsevier]

Abstract: We consider a problem related to the submodular set cover on polymatroids, when the ground set is the family of independent sets of a matroid. The achievement here is getting a strongly polynomial running time with respect to the ground set of the matroid even though the family of independent sets has exponential size. We also address the optimization problem of the maximization of submodular set functions on the independent sets of a matroid. [Copyright &y& Elsevier]

Abstract: Efficient data gathering is an important challenge in sensor networks. In this paper we address the problem of gathering sensed data to the sink of a sensor network minimizing the time to complete the process. We present optimal time data gathering algorithms for any sensor network topology, in the half-duplex with directional antennas model, when each sensor has one data packet to be gathered and merging of packets is not allowed at intermediate nodes. Our results improve on existing approximation algorithms. We approach the gathering problem by obtaining optimal solutions to a path coloring problem in graphs. [Copyright &y& Elsevier]

In this correspondence, we prove that the probability distribution induced on the leaves of a Tunstall parse tree for a given source is a (unique) lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves, and ordered according to the majorization partial order. We apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin. [ABSTRACT FROM AUTHOR]

Abstract: Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost when D is known and if D is unknown, where and are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost , if the size n of the ring is known, and of cost , if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost if the topology of the graph and the initial positions are known to agents. [Copyright &y& Elsevier]

In this paper we study the effect that delayed answers can have on binary search procedures. [Copyright &y& Elsevier]

Motivated by a problem in the design of optical networks, we ask when a graph has a spanning spider (subdivision of a star), or, more generally, a spanning tree with a bounded number of branch vertices. We investigate the existence of these spanning subgraphs in analogy to classical studies of Hamiltonicity. [Copyright &y& Elsevier]

Focuses on a study that considered several algorithmic questions related to the classical problem of simulating the outcomes of a random variable using a limited number of biased coins. Simulating a die with two coins; Simulating a die in optimal average time; Generating arbitrary dice; Conclusion.

Given a network represented by a weighted directed graph G , we consider the problem of finding a bounded cost set of nodes S such that the influence spreading from S in G , within a given time bound, is as large as possible. The dynamics that governs the spread of influence is the following: initially only elements in S are influenced; subsequently at each round, the set of influenced elements is augmented by all nodes in the network that have a sufficiently large number of already influenced neighbors. We prove that the problem is NP-hard, even in simple networks like complete graphs and trees. We also derive a series of positive results. We present exact pseudo-polynomial time algorithms for general trees, that become polynomial time in case the trees are unweighted. This last result improves on previously published results. We also design polynomial time algorithms for general weighted paths and cycles, and for unweighted complete graphs. [ABSTRACT FROM AUTHOR]

Abstract: Motivated by applications in sociology, economy and medicine, we study variants of the Target Set Selection problem, first proposed by Kempe, Kleinberg and Tardos. In our scenario one is given a graph , integer values for each vertex v (thresholds), and the objective is to determine a small set of vertices (target set) that activates a given number (or a given subset) of vertices of G within a prescribed number of rounds. The activation process in G proceeds as follows: initially, at round 0, all vertices in the target set are activated; subsequently at each round every vertex of G becomes activated if at least of its neighbors are already active by round . It is known that the problem of finding a minimum cardinality Target Set that eventually activates the whole graph G is hard to approximate to a factor better than . In this paper we give exact polynomial time algorithms to find minimum cardinality Target Sets in graphs of bounded clique-width, and exact linear time algorithms for trees. [Copyright &y& Elsevier]

We define a family of Distributed Hash Table systems whose aim is to combine the routing efficiency of randomized networks—e.g. optimal average path length O(log 2 n/ δlog δ) with δ degree—with the programmability and startup efficiency of a uniform overlay—that is, a deterministic system in which the overlay network is transitive and greedy routing is optimal. It is known that Ω(log n) is a lower bound on the average path length for uniform overlays with O(log n) degree (Xu et al., IEEE J. Sel. Areas Commun. 22(1), 151–163, ). Our work is inspired by neighbor-of-neighbor (NoN) routing, a recently introduced variation of greedy routing that allows us to achieve optimal average path length in randomized networks. The advantage of our proposal is that of allowing the NoN technique to be implemented without adding any overhead to the corresponding deterministic network. We propose a family of networks parameterized with a positive integer c which measures the amount of randomness that is used. By varying the value c, the system goes from the deterministic case ( c=1) to an “ almost uniform” system. Increasing c to relatively low values allows for routing with asymptotically optimal average path length while retaining most of the advantages of a uniform system, such as easy programmability and quick bootstrap of the nodes entering the system. We also provide a matching lower bound for the average path length of the routing schemes for any c. [ABSTRACT FROM AUTHOR]