Establishing symmetric connectivity in directional wireless sensor networks equipped with $$2\pi /3$$ antennas

In this paper, we study the antenna orientation problem concerning symmetric connectivity in directional wireless sensor networks. We are given a set of nodes each of which is equipped with one directional antenna with beam-width \(\theta = 2\pi /3\) and is initially assigned a transmission range 1 that yields a connected unit disk graph spanning all nodes. The objective of the problem is to compute an orientation of the antennas and to find a minimum transmission power range \(r=O(1)\) such that the induced symmetric communication graph is connected. We propose two algorithms that orient the antennas to yield symmetric connected communication graphs where the transmission power ranges are bounded by 6 and 5, which are currently the best results for this problem. We also study the performance of our algorithms through simulations.     

Algorithms for the minimum non-separating path and the balanced connected bipartition problems on grid graphs

For given a pair of nodes in a graph, the minimum non-separating path problem looks for a minimum weight path between the two nodes such that the remaining graph after removing the path is still connected. The balanced connected bipartition (BCP₂) problem looks for a way to bipartition a graph into two connected subgraphs with their weights as equal as possible. In this paper we present an algorithm in time O(NlogN) for finding a minimum weight non-separating path between two given nodes in a grid graph of N nodes with positive weight. This result leads to a 5/4-approximation algorithm for the BCP₂ problem on grid graphs, which is the currently best ratio achieved in polynomial time. We also developed an exact algorithm for the BCP₂ problem on grid graphs. Based on the exact algorithm and a rounding technique, we show an approximation scheme, which is a fully polynomial time approximation scheme for fixed number of rows.     

OMEGa: an optimistic most energy gain method for minimum energy multicasting in wireless ad hoc networks

In wireless ad hoc networks where every device runs on its own battery, the energy consumption is critical to lengthen the network lifetime. The communication among devices in the network can be categorized as unicasting and multicasting (including broadcasting). For the case of unicasting, computing the energy optimal path between the two communicating nodes is polynomially solvable by computing the shortest path. But for the case of multicasting, shortest path or minimum spanning tree does not guarantee an energy optimal communication. In this paper, we present our novel approach, Optimistic Most Energy Gain (OMEGa) method, for the minimum energy multicasting in wireless ad hoc networks. OMEGa aims at maximum utilization of Wireless Multicast Advantage (WMA), which essentially means covering more nodes by using larger energy. Both theoretical and experimental analysis shows OMEGa method performs very well. Research is partially supported by NSF and Air Force grants.     

On the complexity of path problems in properly colored directed graphs

We address the complexity class of several problems related to finding a path in a properly colored directed graph. A properly colored graph is defined as a graph G whose vertex set is partitioned into $\mathcal{X}(G)$ stable subsets, where $\mathcal{X}(G)$ denotes the chromatic number of G. We show that to find a simple path that meets all the colors in a properly colored directed graph is NP-complete, and so are the problems of finding a shortest and longest of such paths between two specific nodes.     

On dual power assignment optimization for biconnectivity

Topology control is an important technology of wireless ad hoc networks to achieve energy efficiency and fault tolerance. In this paper, we study the dual power assignment problem for 2-edge connectivity and 2-vertex connectivity in the symmetric graphical model which is a combinatorial optimization problem from topology control technology.The problem is arisen from the following origin. In a wireless ad hoc network where each node can switch its transmission power between high-level and low-level, how can we establish a fault-tolerantly connected network topology in the most energy-efficient way? Specifically, the objective is to minimize the number of nodes assigned with high power and yet achieve 2-edge connectivity or 2-vertex connectivity.We addressed these optimization problems (2-edge connectivity and 2-vertex connectivity version) under the general graph model in (Wang et al. in Theor. Comput. Sci., 2008). In this paper, we propose a novel approximation algorithm, called Candidate Set Filtering algorithm, to compute nearly-optimal solutions. Specifically, our algorithm can achieve 3.67-approximation ratio for both 2-edge connectivity and 2-vertex connectivity, which improves the existing 4-approximation algorithms for these two cases.     

Online tree node assignment with resource augmentation

Given a complete binary tree of height h, the online tree node assignment problem is to serve a sequence of assignment/release requests, where an assignment request, with an integer parameter 0≤i≤h, is served by assigning a (tree) node of level (or height) i and a release request is served by releasing a specified assigned node. The node assignments have to guarantee that no node is assigned to two assignment requests unreleased, and every leaf-to-root path of the tree contains at most one assigned node. With assigned node reassignments allowed, the target of the problem is to minimize the number of assignments/reassignments, i.e., the cost, to serve the whole sequence of requests. This online tree node assignment problem is fundamental to many applications, including OVSF code assignment in WCDMA networks, buddy memory allocation and hypercube subcube allocation.     

A Genetic Algorithm for the Weight Setting Problem in OSPF Routing

With the growth of the Internet, Internet Service Providers (ISPs) try to meet the increasing traffic demand with new technology and improved utilization of existing resources. Routing of data packets can affect network utilization. Packets are sent along network paths from source to destination following a protocol. Open Shortest Path First (OSPF) is the most commonly used intra-domain Internet routing protocol (IRP). Traffic flow is routed along shortest paths, splitting flow at nodes with several outgoing links on a shortest path to the destination IP address. Link weights are assigned by the network operator. A path length is the sum of the weights of the links in the path. The OSPF weight setting (OSPFWS) problem seeks a set of weights that optimizes network performance. We study the problem of optimizing OSPF weights, given a set of projected demands, with the objective of minimizing network congestion. The weight assignment problem is NP-hard. We present a genetic algorithm (GA) to solve the OSPFWS problem. We compare our results with the best known and commonly used heuristics for OSPF weight setting, as well as with a lower bound of the optimal multi-commodity flow routing, which is a linear programming relaxation of the OSPFWS problem. Computational experiments are made on the AT&T Worldnet backbone with projected demands, and on twelve instances of synthetic networks.     

Reliable Ad Hoc Routing Based on Mobility Prediction

Reliability is a very important issue in Mobile Ad hoc NETworks (MANETs). Shortest paths are usually used to route packets in MANETs. However, a shortest path may fail quickly, because some of the wireless links along a shortest path may be broken shortly after the path is established due to mobility of mobile nodes. Rediscovering routes may result in substantial data loss and message exchange overhead. In this paper, we study reliable ad hoc routing in the urban environment. Specifically, we formulate and study two optimization problems. In the minimum Cost Duration-bounded Path (CDP) routing problem, we seek a minimum cost source to destination path with duration no less than a given threshold. In the maximum Duration Cost-bounded Path (DCP) routing problem, we seek a maximum duration source to destination path with cost no greater than a given threshold. We use a waypoint graph to model the working area of a MANET and present an offline algorithm to compute a duration prediction table for the given waypoint graph. An entry in the duration prediction table contains the guaranteed worst-case duration of the corresponding wireless link. We then present an efficient algorithm which computes a minimum cost duration-bounded path, using the information provided in the duration prediction table. We also present a heuristic algorithm for the DCP routing problem. In addition, we show that the proposed prediction and routing schemes can be easily applied for designing reliable ad hoc routing protocols. Simulation results show that our mobility prediction based routing algorithms lead to higher network throughput and longer average path duration, compared with the shortest path routing. This research was supported in part by ARO grant W911NF-04-1-0385 and NSF grant CCF-0431167. The information reported here does not reflect the position or the policy of the federal government.     

On optimal placement of relay nodes for reliable connectivity in wireless sensor networks

The paper addresses the relay node placement problem in two-tiered wireless sensor networks. Given a set of sensor nodes in Euclidean plane, our objective is to place minimum number of relay nodes to forward data packets from sensor nodes to the sink, such that: 1) the network is connected, 2) the network is 2-connected. For case one, we propose a (6+ε)-approximation algorithm for any ε > 0 with polynomial running time when ε is fixed. For case two, we propose two approximation algorithms with (24+ε) and (6/T+12+ε), respectively, where T is the ratio of the number of relay nodes placed in case one to the number of sensors. We further extend the results to the cases where communication radiuses of sensor nodes and relay nodes are different from each other.     

Graph Algorithms with Small Communication Costs

We study minimizing communication cost in parallel algorithm design, by minimizing the number of communication phases in coarse-grained parallel computers. There have been several recent papers dealing with parallel algorithms of small communication cost under different models. Most of these results are for computational geometry problems. For these problems it has been possible to decompose tasks into appropriate subproblems in a communication-efficient way. It appears to be somewhat more difficult to design parallel algorithms with small communication phases for graph theory problems. In this paper we focus on the design of deterministic algorithms with a small number of communication phases for the list ranking problem and the shortest path problem.     

On broadcasting in unicyclic graphs

Broadcasting is an information dissemination problem in a connected network, in which one node, called the originator, disseminates a message to all other nodes by placing a series of calls along the communication lines of the network. Finding the broadcast time of a vertex in an arbitrary graph is NP-complete. The polynomial time solvability is shown only for trees. In this paper we present a linear algorithm that determines the broadcast time of any originator in an arbitrary unicyclic graph. As a byproduct, we find a broadcast center of the unicyclic graph. We also present an O(|V|+k ²) algorithm to find the broadcast time of an arbitrary unicyclic graph, where k is the length of the cycle. In the last section we give tight lower and upper bounds on broadcast time of a spanning tree based on the broadcast time of the unicyclic graph. The results of Sects. 2, 3 and most of the proofs in Sects. 2, 3 of this paper are presented by Harutyunyan and Maraachlian (Proceedings of 13th annual COCOON, pp. 372–383, 2007). All results in Sects. 4, 5 and the complete proof of Theorem 3 are new results.     

Packing trees in communication networks

Given an undirected edge-capacitated graph and given (possibly) different subsets of vertices, we consider the problem of selecting a maximum (weighted) set of Steiner trees, each tree spanning a subset of vertices, without violating the capacity constraints. This problem is motivated by applications in multicast communication networks. We give an integer linear programming (ILP) formulation for the problem, and observe that its linear programming (LP) relaxation is a fractional packing problem with exponentially many variables and a block (sub-)problem that cannot be solved in polynomial time. To this end, we take an r-approximate block solver (a weak block solver) to develop a (1−ε)/r-approximation algorithm for the LP relaxation. The algorithm has a polynomial coordination complexity for any ε∈(0,1). To the best of our knowledge, this is the first approximation result for fractional packing problems with only weak block solvers (with arbitrarily large approximation ratio) and a coordination complexity that is polynomial in the input size. This leads also to an approximation algorithm for the underlying tree packing problem. Finally, we extend our results to an important multicast routing and wavelength assignment problem in optical networks, where each Steiner tree is to be assigned one of a limited set of given wavelengths, so that trees crossing the same fiber are assigned different wavelengths. A preliminary version of this paper appeared in the Proceedings of the 1st Workshop on Internet and Network Economics (WINE 2005), LNCS, vol. 3828, pp. 688–697. Research supported by a MITACS grant for all the authors, an NSERC post doctoral fellowship for the first author, the NSERC Discovery Grant #5-48923 for the second and fourth author, NSERC Discovery Grant #15296 for the third author, the Canada Research Chair Program for the second author, and an NSERC industrial and development fellowship for the fourth author.     

b-Tree Facets for the Simple Graph Partitioning Polytope

The simple graph partitioning problem is to partition an edge-weighted graph into mutually disjoint subgraphs, each consisting of no more than b nodes, such that the sum of the weights of all edges in the subgraphs is maximal. In this paper we introduce a large class of facet defining inequalities for the simple graph partitioning polytopes _n (b), b 3, associated with the complete graph on n nodes. These inequalities are induced by a graph configuration which is built upon trees of cardinality b. We provide a closed-form theorem that states all necessary and sufficient conditions for the facet defining property of the inequalities.     

Network Reconfiguration for Increasing Transportation System Resilience Under Extreme Events

Evacuating residents out of affected areas is an important strategy for mitigating the impact of natural disasters. However, the resulting abrupt increase in the travel demand during evacuation causes severe congestions across the transportation system, which thereby interrupts other commuters' regular activities. In this article, a bilevel mathematical optimization model is formulated to address this issue, and our research objective is to maximize the transportation system resilience and restore its performance through two network reconfiguration schemes: contraflow (also referred to as lane reversal) and crossing elimination at intersections. Mathematical models are developed to represent the two reconfiguration schemes and characterize the interactions between traffic operators and passengers. Specifically, traffic operators act as leaders to determine the optimal system reconfiguration to minimize the total travel time for all the users (both evacuees and regular commuters), while passengers act as followers by freely choosing the path with the minimum travel time, which eventually converges to a user equilibrium state. For each given network reconfiguration, the lower‐level problem is formulated as a traffic assignment problem (TAP) where each user tries to minimize his/her own travel time. To tackle the lower‐level optimization problem, a gradient projection method is leveraged to shift the flow from other nonshortest paths to the shortest path between each origin–destination pair, eventually converging to the user equilibrium traffic assignment. The upper‐level problem is formulated as a constrained discrete optimization problem, and a probabilistic solution discovery algorithm is used to obtain the near‐optimal solution. Two numerical examples are used to demonstrate the effectiveness of the proposed method in restoring the traffic system performance.     

Finding the anti-block vital edge of a shortest path between two nodes

Let P _G (s,t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u∈P _G (s,t)=(s,…,u,v,…,t) is defined as a shortest path P _G−e (u,t) from u to t which does not make use of (u,v). In this paper, we focus on the problem of finding an edge e=(u,v)∈P _G (s,t) whose removal produces a detour at node u such that the ratio of the length of P _G−e (u,t) to the length of P _G (u,t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown. This research is supported by NSF of China under Grants 70471035, 70525004, 701210001 and 60736027, and PSF of China under Grant 20060401003.     

Data relaying with constraints in hierarchical sensor networks

In this paper we present and discuss several optimisation problems that arise in the management of data flow in wireless sensor networks (WSNets). We consider a hierarchical architecture for WSNets composed of sensors, relays, and relay gateways. Sensors send data they generate at a known average bit rate to relays in one hop. The relay nodes use a multi-hop mechanism to reach a set of assigned gateways which then forward the data directly to the base station. We are concerned with finding an assignment of relay gateways to relays so that certain constraints are satisfied. We define a unified model in which constraints such as lifetime, data delay, and data flow splitting are formulated in terms of four optimisation problem in graphs.     

Strong minimum energy hierarchical topology in wireless sensor networks

Given a set of \(n\) sensors, the strong minimum energy topology (SMET) problem in a wireless sensor network is to assign transmit powers to all sensors such that (i) the graph induced only using the bi-directional links is connected, that is, there is a path between every pair of sensors, and (ii) the sum of the transmit powers of all the sensors is minimum. This problem is known to be NP-hard. In this paper, we study a special case of the SMET problem, namely , the \(k\)-strong minimum energy hierarchical topology (\(k\)-SMEHT) problem. Given a set of \(n\) sensors and an integer \(k\), the \(k\)-SMEHT problem is to assign transmission powers to all sensors such that (i) the graph induced using only bi-directional links is connected, (ii) at most \(k\) nodes of the graph induced using only bi-directional links have two or more neighbors, that is they are non-pendant nodes, and (iii) the sum of the transmit powers of all the sensors in \(G\) is minimum. We show that \(k\)-SMEHT problem is NP-hard for arbitrary \(k\). However, we propose a \(\frac{k+1}{2}\)-approximation algorithm for \(k\)-SMEHT problem, when \(k\) is a fixed constant. Finally, we propose a polynomial time algorithm for the \(k\)-SMEHT problem for \(k=2\).     

Relay node placement in two-tiered wireless sensor networks with base stations

This paper addresses the relay node placement problem in two-tiered wireless sensor networks with base stations, which aims to deploy a minimum number of relay nodes to achieve certain coverage and connectivity requirement. Under the assumption that the communication range of the sensor nodes is no more than that of the relay node, we present a polynomial time (5+?)-approximation algorithm for the 1-coverage 1-connected problem. Furthermore, we consider the fault tolerant problem in the network, we present a polynomial time (20+?)-approximation algorithm for the 2-coverage 2-connected problem, where ? is any given positive constant. For the k-coverage 2-connected situation, we present a polynomial time (15k?10+?)-approximation algorithm.     

On Approximating a Scheduling Problem

Given a set of communication tasks (best described in terms of a weighted bipartite graph where one set of nodes denotes the senders, the other set the receivers, edges are communication tasks, and the weight of an edge is the time required for transmission), we wish to minimize the total time required for the completion of all communication tasks assuming that tasks can be preempted (that is, each edge can be subdivided into many edges with weights adding up to the edge's original weight) and that preemption comes with a cost. In this paper, we first prove that one cannot approximate this problem within a factor smaller than unless P=NP. It is known that a simple approximation algorithm achieves within a ratio of two (H. Choi and S.L. Hakimi, Algorithmica, vol. 3, pp. 223–245, 1988). However, our experimental results show that its performance is worse than the originally proposed heuristic algorithm (I.S. Gopal and C.K. Wong, IEEE Transactions on Communications, vol. 33, pp. 497–501, 1985). We devise a more sophisticated algorithm, called the potential function algorithm which, on the one hand, achieves a provable approximation ratio of two, and on the other hand, shows very good experimental performance. Moreover, the way in which our more sophisticated algorithm derives from the simple one, suggests a hierarchy of algorithms, all of which have a worst-case performance at most two, but which we suspect to have increasingly better performance, both in worst case and with actual instances.     

The planning of cycle trips in the province of East Flanders

Traditional route planners assist in finding the shortest or fastest route from one place to another. This paper presents a novel approach to path finding in a directed graph, namely a target distance, motivated by the problem that a recreational cyclist deals with when searching a nice route of a certain length. The problem is defined as a variant of the arc orienteering problem (AOP), a new combinatorial optimisation problem in which the score of a route in a directed graph has to be maximised by visiting arcs, while each arc can be visited at most once and the total cost of the route should not exceed a predefined cost. The contribution of this paper is threefold: (1) a mathematical model of the AOP is provided, (2) a metaheuristic method that solves AOP instances to near optimality in 1 s of execution time, is proposed and evaluated, and (3) two real-life applications of the method are presented. An on-line cycle route planning application offers personalised cycle routes based on user preferences, and an SMS service provides cyclists “in the field” with routes on demand.