Approximating Minimum Independent Dominating Sets in Wireless Networks

We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in a graph that is also independent. It thus combines the advantages of both structures, and there are many applications that rely on these two structures e.g. in the area of wireless ad hoc networks. The presented approach yields a robust algorithm, that is, the algorithm accepts any undirected graph as input, and returns a (1+")- pproximate minimum dominating set, or a certificate showing that the input graph does not reflect a wireless network

Local Approximation Schemes for Ad Hoc and Sensor Networks

We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs

A Distributed and Self-Organizing Scheduling Algorithm for Energy-Efficient Data Aggregation in Wireless Sensor Networks

Wireless sensor networks (WSNs) are increasingly being used to monitor various parameters in a wide range of environmental monitoring applications. In many instances, environmental scientists are interested in collecting raw data using long-running queries injected into a WSN for analyzing at a later stage, rather than injecting snap-shot queries containing data-reducing operators (e.g., MIN, MAX, AVG) that aggregate data. Collection of raw data poses a challenge to WSNs as very large amounts of data need to be transported through the network. This not only leads to high levels of energy consumption and thus diminished network lifetime but also results in poor data quality as much of the data may be lost due to the limited bandwidth of present-day sensor nodes. We alleviate this problem by allowing certain nodes in the network to aggregate data by taking advantage of spatial and temporal correlations of various physical parameters and thus eliminating the transmission of redundant data. In this article we present a distributed scheduling algorithm that decides when a particular node should perform this novel type of aggregation. The scheduling algorithm autonomously reassigns schedules when changes in network topology, due to failing or newly added nodes, are detected. Such changes in topology are detected using cross-layer information from the underlying MAC layer. We first present the theoretical performance bounds of our algorithm. We then present simulation results, which indicate a reduction in message transmissions of up to 85% and an increase in network lifetime of up to 92% when compared to collecting raw data. Our algorithm is also capable of completely eliminating dropped messages caused by buffer overflow

A theorem on the absence of phase transitions in one-dimensional growth models with onsite periodic potentials

We rigorously prove that a wide class of one-dimensional growth models with onsite periodic potential, such as the discrete sine-Gordon model, have no phase transition at any temperature $T>0$. The proof relies on the spectral analysis of the transfer operator associated to the models. We show that this operator is Hilbert-Schmidt and that its maximum eigenvalue is an analytic function of temperature.Comment: 6 pages, no figures, submitted to J Phys A: Math Ge

A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a {\em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within (1+\eps) ratio of the optimum if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG's is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an O(\frac{\log^* n}{\eps^{O(1)}}) time distributed PTAS. We consider a weighted version of the clique partition problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet, surprisingly, it admits a (2+\eps)-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the {\em unweighted} case for UDGs expressed in standard form.Comment: 21 pages, 9 figure

On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $\alpha$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $\alpha = 1$ by Ding et al. For any constant $\alpha > 1$, we give an $O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})$-approximation algorithm. When $\alpha \geq 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $\alpha =2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $\epsilon > 0$, the problem admits no polynomial-time $2^{\log^{1-\epsilon}n}$-approximation algorithm, improving upon the $\Omega(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption)

Interview investigation of insecure attachment styles as mediators between poor childhood care and schizophrenia-spectrum phenomenology

Background Insecure attachment styles have received theoretical attention and some initial empirical support as mediators between childhood adverse experiences and psychotic phenomena; however, further specificity needs investigating. The present interview study aimed to examine (i) whether two forms of poor childhood care, namely parental antipathy and role reversal, were associated with subclinical positive and negative symptoms and schizophrenia-spectrum personality disorder (PD) traits, and (ii) whether such associations were mediated by specific insecure attachment styles. Method A total of 214 nonclinical young adults were interviewed for subclinical symptoms (Comprehensive Assessment of At-Risk Mental States), schizophrenia-spectrum PDs (Structured Clinical Interview for DSM-IV Axis II Disorders), poor childhood care (Childhood Experience of Care and Abuse Interview), and attachment style (Attachment Style Interview). Participants also completed the Beck Depression Inventory-II and all the analyses were conducted partialling out the effects of depressive symptoms. Results Both parental antipathy and role reversal were associated with subclinical positive symptoms and with paranoid and schizotypal PD traits. Role reversal was also associated with subclinical negative symptoms. Angry-dismissive attachment mediated associations between antipathy and subclinical positive symptoms and both angry-dismissive and enmeshed attachment mediated associations of antipathy with paranoid and schizotypal PD traits. Enmeshed attachment mediated associations of role reversal with paranoid and schizotypal PD traits. Conclusions Attachment theory can inform lifespan models of how adverse developmental environments may increase the risk for psychosis. Insecure attachment provides a promising mechanism for understanding the development of schizophrenia-spectrum phenomenology and may offer a useful target for prophylactic intervention

Independent and Dominating Sets in Wireless Communication Graphs

Wireless ad hoc networks are advancing rapidly, both in research and more and more into our everyday lives. Wireless sensor networks are a prime example of a new technology that has gained a lot of attention in the literature, and that is going to enhance the way we view and interact with the environment. These ad hoc and sensor networks are modeled by communication graphs which give the communication links between some devices or nodes equipped with wireless transceivers or radios. The nature of wireless transmissions does not lead to an arbitrary network topology, but creates a network with certain properties. These properties include the important structure of bounded growth. In this thesis, we look at the structures and resulting optimization problems of independent and dominating sets in on graphs that model wireless communication networks. Independent and dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. In a communication graph, an independent set consists of vertices that cannot communicate with one another directly. Such a set is commonly used in clustering strategies, e.g. to obtain a hierarchical view of the network. A dominating set is given by a set of vertices so that every vertex in the graph is either in this set, or adjacent to a vertex from this set. Dominating sets of small cardinality are frequently used for backbone structures in communication networks, e.g. to obtain efficient multi-hop routing protocols. For the optimization problems of seeking independent sets of large cardinality or weight (Maximum Independent Set problem), and seeking dominating sets of small cardinality (Minimum Dominating Set problem) on graphs of polynomially bounded growth, we present and discuss polynomial-time approximation schemes (PTAS). The algorithms presented are robust in the sense that they accept an undirected graph, and return a desired solution or a certificate that shows that the instance does not satisfy the structural properties of a wireless communication graph. Wireless ad hoc and sensor networks usually lack a central control instance. Local, distributed algorithms are designed to operate in such a scenario. We propose a fast local algorithm that constructs a so-called maximal independent set, which is also dominating. We also extend the ideas behind the centrally executed PTAS towards a local approach. This gives a distributed approximation scheme for the Maximum Independent Set and Minimum Dominating Set problems on wireless communication graphs. The second part of the thesis is devoted to an application of independent and dominating sets in a wireless sensor network. We present the design of an energy-efficient communication strategy for low-resource, large-scale wireless networks that is based on an integrated approach stemming over various layers of the communication stack. The transmission of data packets in this scheme is done in a scheduled manner, thus reducing the number of collisions due to simultaneous transmissions. The approach, called EMACs, also includes the creation and maintenance of a connected backbone in the network that is used for implicit sleep-state scheduling of the nodes to conserve additional energy. Some results obtained from a practical implementation of this scheme indicate that the EMACs communication scheme improves the network lifetime compared to existing schemes for wireless sensor networks. This thesis answers and contributes to several open questions from the literature. The characterization of wireless communication graphs by the bounded growth property includes geometrically defined Unit Disk Graphs. By giving a PTAS that does not exploit geometric information for the Maximum Independent and Minimum Dominating Set problems on these graphs, we give a positive answer to the open problem of the existence of a PTAS for Unit Disk Graphs without geometric representation. Local, distributed maximal independent set computation has received a lot of attention in the literature, and a fast, i.e. poly-logarithmic distributed time approach is a longstanding open problem for communication networks in general. At least for wireless communication networks, we present the first such fast, local approach

Approximation schemes for wireless networks

Wireless networks are created by the communication links between a collection of radio transceivers. The nature of wireless transmissions does not lead to arbitrary undirected graphs but to structured graphs which we characterize by the polynomially bounded growth property. In contrast to many existing graph models for wireless networks, the property of polynomially bounded growth is defined independently of geometric data such as positional information. On such wireless networks, we present an approach that can be used to create polynomial-time approximation schemes for several optimization problems called the local neighborhood-based scheme. We apply this approach to the problems of seeking maximum (weight) independent sets and minimum dominating sets. These are two important problems in the area of wireless communication networks and are also used in many applications ranging from clustering to routing strategies. However, the approach is presented in a general fashion since it can be applied to other problems as well. The approach for the approximation schemes is robust in the sense that it accepts any undirected graph as input and either outputs a solution of desired quality or correctly asserts that the graph presented as input does not satisfy the structural assumption of a wireless network (an NP-hard problem)