Optimization in Geometric Graphs: Complexity and Approximation

We consider several related problems arising in geometric graphs. In particular, we investigate the computational complexity and approximability properties of several optimization problems in unit ball graphs and develop algorithms to find exact and approximate solutions. In addition, we establish complexity-based theoretical justifications for several greedy heuristics. Unit ball graphs, which are defined in the three dimensional Euclidian space, have several application areas such as computational geometry, facility location and, particularly, wireless communication networks. Efficient operation of wireless networks involves several decision problems that can be reduced to well known optimization problems in graph theory. For instance, the notion of a \virtual backbone" in a wire- less network is strongly related to a minimum connected dominating set in its graph theoretic representation. Motivated by the vastness of application areas, we study several problems including maximum independent set, minimum vertex coloring, minimum clique partition, max-cut and min-bisection. Although these problems have been widely studied in the context of unit disk graphs, which are the two dimensional version of unit ball graphs, there is no established result on the complexity and approximation status for some of them in unit ball graphs. Furthermore, unit ball graphs can provide a better representation of real networks since the nodes are deployed in the three dimensional space. We prove complexity results and propose solution procedures for several problems using geometrical properties of these graphs. We outline a matching-based branch and bound solution procedure for the maximum k-clique problem in unit disk graphs and demonstrate its effectiveness through computational tests. We propose using minimum bottleneck connected dominating set problem in order to determine the optimal transmission range of a wireless network that will ensure a certain size of "virtual backbone". We prove that this problem is NP-hard in general graphs but solvable in polynomial time in unit disk and unit ball graphs. We also demonstrate work on theoretical foundations for simple greedy heuristics. Particularly, similar to the notion of "best" approximation algorithms with respect to their approximation ratios, we prove that several simple greedy heuristics are "best" in the sense that it is NP-hard to recognize the gap between the greedy solution and the optimal solution. We show results for several well known problems such as maximum clique, maximum independent set, minimum vertex coloring and discuss extensions of these results to a more general class of problems. In addition, we propose a "worst-out" heuristic based on edge contractions for the max-cut problem and provide analytical and experimental comparisons with a well known "best-in" approach and its modified versions

Heuristics for Network Coding in Wireless Networks

Multicast is a central challenge for emerging multi-hop wireless architectures such as wireless mesh networks, because of its substantial cost in terms of bandwidth. In this report, we study one specific case of multicast: broadcasting, sending data from one source to all nodes, in a multi-hop wireless network. The broadcast we focus on is based on network coding, a promising avenue for reducing cost; previous work of ours showed that the performance of network coding with simple heuristics is asymptotically optimal: each transmission is beneficial to nearly every receiver. This is for homogenous and large networks of the plan. But for small, sparse or for inhomogeneous networks, some additional heuristics are required. This report proposes such additional new heuristics (for selecting rates) for broadcasting with network coding. Our heuristics are intended to use only simple local topology information. We detail the logic of the heuristics, and with experimental results, we illustrate the behavior of the heuristics, and demonstrate their excellent performance

An Order-based Algorithm for Minimum Dominating Set with Application in Graph Mining

Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS$_o$) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLS$_o$ performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLS$_o$ that is suitable for solving the minimum weight dominating set problem. The application of RLS$_o$ in graph mining is also briefly demonstrated

Maximizing the Probability of Delivery of Multipoint Relay Broadcast Protocol in Wireless Ad Hoc Networks with a Realistic Physical Layer

It is now commonly accepted that the unit disk graph used to model the physical layer in wireless networks does not reflect real radio transmissions, and that the lognormal shadowing model better suits to experimental simulations. Previous work on realistic scenarios focused on unicast, while broadcast requirements are fundamentally different and cannot be derived from unicast case. Therefore, broadcast protocols must be adapted in order to still be efficient under realistic assumptions. In this paper, we study the well-known multipoint relay protocol (MPR). In the latter, each node has to choose a set of neighbors to act as relays in order to cover the whole 2-hop neighborhood. We give experimental results showing that the original method provided to select the set of relays does not give good results with the realistic model. We also provide three new heuristics in replacement and their performances which demonstrate that they better suit to the considered model. The first one maximizes the probability of correct reception between the node and the considered relays multiplied by their coverage in the 2-hop neighborhood. The second one replaces the coverage by the average of the probabilities of correct reception between the considered neighbor and the 2-hop neighbors it covers. Finally, the third heuristic keeps the same concept as the second one, but tries to maximize the coverage level of the 2-hop neighborhood: 2-hop neighbors are still being considered as uncovered while their coverage level is not higher than a given coverage threshold, many neighbors may thus be selected to cover the same 2-hop neighbors

GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool

Efficient Algorithms for Distributed Detection of Holes and Boundaries in Wireless Networks

We propose two novel algorithms for distributed and location-free boundary recognition in wireless sensor networks. Both approaches enable a node to decide autonomously whether it is a boundary node, based solely on connectivity information of a small neighborhood. This makes our algorithms highly applicable for dynamic networks where nodes can move or become inoperative. We compare our algorithms qualitatively and quantitatively with several previous approaches. In extensive simulations, we consider various models and scenarios. Although our algorithms use less information than most other approaches, they produce significantly better results. They are very robust against variations in node degree and do not rely on simplified assumptions of the communication model. Moreover, they are much easier to implement on real sensor nodes than most existing approaches.Comment: extended version of accepted submission to SEA 201

A PTAS for the minimum dominating set problem in unit disk graphs

We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. \u