Showing papers on "Equal-cost multi-path routing published in 2018"

Segment Routing Architecture

Abstract: Segment Routing (SR) leverages the source routing paradigm. A node steers a packet through an ordered list of instructions, called segments. A segment can represent any instruction, topological or service-based. A segment can have a semantic local to an SR node or global within an SR domain. SR allows to enforce a flow through any topological path while maintaining per-flow state only at the ingress nodes to the SR domain. Segment Routing can be directly applied to the MPLS architecture with no change on the forwarding plane. A segment is encoded as an MPLS label. An ordered list of segments is encoded as a stack of labels. The segment to process is on the top of the stack. Upon completion of a segment, the related label is popped from the stack. Segment Routing can be applied to the IPv6 architecture, with a new type of routing header. A segment is encoded as an IPv6 address. An ordered list of segments is encoded as an ordered list of IPv6 addresses in the routing header. The active segment is indicated by the Destination Address of the packet. The next active segment is indicated by a pointer in the new routing header.

Energy-Aware Dual-Path Geographic Routing to Bypass Routing Holes in Wireless Sensor Networks

Abstract: Geographic routing has been considered as an attractive approach for resource-constrained wireless sensor networks (WSNs) since it exploits local location information instead of global topology information to route data. However, this routing approach often suffers from the routing hole (i.e., an area free of nodes in the direction closer to destination) in various environments such as buildings and obstacles during data delivery, resulting in route failure. Currently, existing geographic routing protocols tend to walk along only one side of the routing holes to recover the route, thus achieving suboptimal network performance such as longer delivery delay and lower delivery ratio. Furthermore, these protocols cannot guarantee that all packets are delivered in an energy-efficient manner once encountering routing holes. In this paper, we focus on addressing these issues and propose an energy-aware dual-path geographic routing (EDGR) protocol for better route recovery from routing holes. EDGR adaptively utilizes the location information, residual energy, and the characteristics of energy consumption to make routing decisions, and dynamically exploits two node-disjoint anchor lists, passing through two sides of the routing holes, to shift routing path for load balance. Moreover, we extend EDGR into three-dimensional (3D) sensor networks to provide energy-aware routing for routing hole detour. Simulation results demonstrate that EDGR exhibits higher energy efficiency, and has moderate performance improvements on network lifetime, packet delivery ratio, and delivery delay, compared to other geographic routing protocols in WSNs over a variety of communication scenarios passing through routing holes. The proposed EDGR is much applicable to resource-constrained WSNs with routing holes.

BERA: a biogeography-based energy saving routing architecture for wireless sensor networks

Abstract: Biogeography-based optimization (BBO) is a relatively new paradigm for optimization which is yet to be explored to solve complex optimization problems to prove its full potential. In wireless sensor networks (WSNs), optimal cluster head selection and routing are two well-known optimization problems. Researchers often use hierarchal cluster-based routing, in which power consumption of cluster heads (CHs) is very high due to its extra functionalities such as receiving and aggregating the data from its member sensor nodes and transmitting the aggregated data to the base station (BS). Therefore, proper care should be taken while selecting the CHs to enhance the life of the network. After formation of the clusters, data to be routed to the BS in inter-cluster fashion for further enhancing the life of WSNs. In this paper, a biogeography-based energy saving routing architecture (BERA) is proposed for CH selection and routing. The biogeography-based CH selection algorithm is proposed with an efficient encoding scheme of a habitat and by formulating a novel fitness function that uses residual energy and distance as its metrics. The BBO-based routing algorithm is also proposed. The efficient encoding scheme of a habitat is developed, and its fitness function considers the node degree in addition to residual energy and distance. To exhibit the performance of BERA, it is extensively tested with some existing routing algorithms such as DHCR, Hybrid routing, EADC and some bio-inspired algorithms, namely GA and PSO. Simulation results confirm the superiority/competitiveness of the proposed algorithm over existing techniques.

A Microbial Inspired Routing Protocol for VANETs

Abstract: We present a bio-inspired unicast routing protocol for vehicular ad hoc networks which uses the cellular attractor selection mechanism to select next hops. The proposed unicast routing protocol based on attractor selecting (URAS) is an opportunistic routing protocol, which is able to change itself adaptively to the complex and dynamic environment by routing feedback packets. We further employ a multiattribute decision-making strategy, the technique for order preference by similarity to an ideal solution, to reduce the number of redundant candidates for next-hop selection, so as to enhance the performance of attractor selection mechanism. Once the routing path is found, URAS maintains the current path or finds another better path adaptively based on the performance of current path, that is, it can self-evolution until the best routing path is found. Our simulation study compares the proposed solution with the state-of-the-art schemes, and shows the robustness and effectiveness of the proposed routing protocol and the significant performance improvement, in terms of packet delivery, end-to-end delay, and congestion, over the conventional method.

Fog computing enabling geographic routing for urban area vehicular network

Abstract: Geographic routing scheme has received considerable attention recently. We present a position-based routing scheme called improved geographic routing (IGR) for the inter-vehicle communication in city environments. IGR uses the vehicular fog computing to make the best utilization of the vehicular communication and computational resources. IGR consists of two modes: (i) junction selection according to the distance to the destination and the vehicle density of each street, and (ii) an improved greedy forwarding strategy to transmit a data packet between two junctions. In the improved greedy forwarding mode, link error rate is considered in the path selection. Simulations are conducted to evaluate the performance of IGR. Simulation results show that IGR has a significant improvement in terms of the achieved packet rate and end-to-end delay.

A survey of recent advances in vehicle routing problems

Abstract: A vehicle routing problem involves finding a set of optimal route for a fleet of capacitated vehicles which are available at a location to service the demands of a set of customers. In its simplest form, a customer is required to be visited once and the capacity of a vehicle must not be exceeded. In this paper, a review of four common and applicable variants of the vehicle routing problem, namely, capacitated vehicle routing problem, vehicle routing problem with time windows, periodic vehicle routing problem and the dynamic vehicle routing problem, are considered based on formulation techniques, methods of solution and areas of application. A summary table is presented for each variant to emphasis some key features that represent direction of current research.

On Caching and Routing in Information-Centric Networks

Abstract: With the exponential growth of content in recent years, users are primarily interested in obtaining particular content and are not concerned with the host housing the content. By treating content as a first class citizen, information- centric networks (ICN) seek to transform the Internet from a host-to-host communication model to a content-centric model. A key component of ICN is to cache content at storage-enabled routers. By caching content at in-network routers, network performance can be improved by delivering content from routers closer to the user and not from the origin servers (content custodians). In this article, we provide an overview of the state-of-the-art cache management and routing policies in ICN. We present a small comparison study of some state-of-the-art algorithms on a real Internet topology to help the reader appreciate how the different strategies compare against one another. Our simulation results demonstrate that it is hard to pick a winner among existing policies. We conclude the article with a discussion of open research questions.

Packet Routing in 3D Nanonetworks: A Lightweight, Linear-path Scheme

Abstract: Packet routing in nanonetworks requires novel approaches, which can cope with the extreme limitations posed by the nano-scale. Highly lossy wireless channels, extremely limited hardware capabilities and non-unique node identifiers are among the restrictions. The present work offers an addressing and routing solution for static 3D nanonetworks that find applications in material monitoring and programmatic property tuning. The addressing process relies on virtual coordinates from multiple, alternative anchor point sets that act as \emph{viewports}. Each viewport offers different address granularity within the network space, and its selection is optimized by a packet sending node using a novel heuristic. Regarding routing, each node can deduce whether it is located on the linear segment connecting the sender to the recipient node. This deduction is made using integer calculations, node-local information and in a stateless manner, minimizing the computational and storage overhead of the proposed scheme. Most importantly, the nodes can regulate the width of the linear path, thus trading energy efficiency (redundant transmissions) for increased path diversity. This trait can enable future adaptive routing schemes. Extensive evaluation via simulations highlights the advantages of the novel scheme over related approaches.

Learning for Robust Routing Based on Stochastic Game in Cognitive Radio Networks

Abstract: This paper studies the problem of spectrum-aware routing in a multi-hop, multi-channel cognitive radio network when malicious nodes in the secondary network attempt to block the path with mixed attacks. Based on the location and time-variant path delay information, we model the path discovery process as a non-cooperative stochastic game. By exploiting the structure of the underlying Markov Decision Process, we decompose the stochastic routing game into a series of stage games. For each stage game, we propose a distributed strategy learning mechanism based on stochastic fictitious play to learn the equilibrium strategies of joint relay-channel selection in the condition of both limited information exchange and potential routing-toward-primary attacks. We also introduce a trustworthiness evaluation mechanism based on a multi-arm bandit process for normal users to avoid relaying to the sink-hole attackers. Simulation results show that without the need of information flooding, the proposed algorithm is efficient in bypassing the malicious nodes with mixed attacks.

A Multicommodity Flow-Based Detailed Router With Efficient Acceleration Techniques

Abstract: Detailed routing is an important stage in very large scale integrated physical design. Due to the extreme scaling of transistor feature size and the complicated design rules, ensuring routing completion without design rule checking (DRC) violations becomes more and more difficult. Studies have shown that the low routing quality partly results from nonoptimal net-ordering nature of traditional sequential methods. The concurrent routing strategy is always based on an NP-hard model, thus is at a disadvantage in runtime. In this paper, we present a novel concurrent detailed routing algorithm that routes all nets simultaneously. Based on the multicommodity flow model, detailed routing problem with complex design rule constraints is formulated as an integer linear programming. Some model simplification heuristics and efficient model solving algorithms are proposed to improve the runtime. Experimental results show that, the proposed algorithms can reduce the DRC violations by 80%, meanwhile can reduce wirelength and via count by 5% and 8% compared with an industry tool. In addition, the proposed algorithm is general that it can be adopted as an incremental detailed router to refine a routing solution, so the number of DRC violations that industry tool cannot fix are further reduced by 27%.

Delay-Optimal Back-Pressure Routing Algorithm for Multihop Wireless Networks

Abstract: Although the back-pressure (BP) algorithm has been proven to be a throughput-optimal policy for traffic scheduling and routing in wireless networks, the optimal control of delays in the BP algorithm remains an open problem, especially for conventional multihop networks. To enhance the delay performance, we proposed an improved BP algorithm called sojourn-time-based BP (STBP) by introducing a novel delay metric called the sojourn time backlog (STB). The STB considers the queue length and accumulated packet delays comprehensively. It provides more pressure to push forward flows suffering from greater delays. Based on this new metric, the calculation of the routing weight for each packet is determined for maximizing the difference of the STB in the routing process. The proposed routing algorithm is robust and distributed, and does not require any prior knowledge of network connections and load conditions. We analyze the network stability with delay considerations and prove the throughput optimality of the STBP in multihop networks. Simulation results reveal that the enhanced algorithm effectively improves the end-to-end delay while ensuring throughput optimality.

Low overhead hybrid geographic-based routing algorithms with smart partial flooding for 3D ad hoc networks

Abstract: One known design for routing algorithms in mobile ad hoc networks is to use flooding. But, these algorithms are usually suffer from high overhead. Another common design is to use the nodes geographic locations to take routing choices. Current geographical routing algorithms usually address the routing environment in 2D space. However, in real life, nodes could be located in 3D space. To benefit from the advantages of both techniques we propose several 3D new routing algorithms that maximize the packets delivery rate and minimize the overhead. Our first set of algorithms (SPF: smart partial flooding) uses the nodes location to do the flooding in the direction of the destination over a sub-graph of the original dense graph. The second set (Progress–SPF) uses geographical routing to progress as much as possible to the destination, if its not possible, SPF is used over a sub-graph extracted locally. The 3rd set (Progress–SPF–Progress) used geographical routing to progress to the destination, if the progress is not possible, SPF is used over a sub-graph for one step only and then the algorithm goes back to the geographical routing. We evaluate our algorithms and compare them with current routing algorithms. The simulation results show a significant improvement in delivery rate up to $$100\%$$ compared to $$70\%$$ and a huge reduction in overall traffic around $$60\%$$ .

Efficient Robust Routing for Single Commodity Network Flows

Abstract: We study single commodity network flows with suitable robustness and efficiency specs. An original use of a maximum entropy problem for distributions on the paths of the graph turns this problem into a steering problem for Markov chains with prescribed initial and final marginals. From a computational standpoint, viewing scheduling this way is especially attractive in light of the existence of an iterative algorithm to compute the solution. This paper builds on the work proposed by Y. Chen et al. , [“Robust transport over networks,” IEEE Transactions on Automatic Control , vol. 62, no. 9, pp. 4675–4682, Sep. 2017] by introducing an index of efficiency of a transportation plan and points, accordingly, to efficient robust transport policies. In developing the theory, we establish two new invariance properties of the solution (called bridge )—an iterated bridge invariance property and the invariance of the most probable paths . These properties, which were tangentially mentioned in our previous work, are fully developed here. We also show that the distribution on paths of the optimal transport policy, which depends on a “ temperature ” parameter, tends to the solution of the “most economical” but possibly less robust optimal mass transport problem as the temperature goes to zero. The relevance of all of these properties for transport over networks is illustrated in an example.

Aspects of Trusted Routing Communication in Smart Networks

Abstract: In this paper, we have exploited the weight trusted routing (WTR) mechanism to detect and eliminate the malicious nodes involved during the routing path formation in smart-home environments where the routing between the communicating entities is performed through the mesh architecture. Further, to provide a secure communication against malicious behavior of nodes, the proposed mechanism uses Dijkstra’s shortest path routing algorithm in which the weights are deliberated using certain parameters such as node-distance, packet-loss and trust value of each node which is computed using social impact theory optimizer. Moreover, we have presented the network performance trade-off caused by secure path formation with conventional method and have proposed the WTR mechanism for eliminating the potential issues such as packet-loss ratio, end-to-end delay and network throughput. The commercial simulator NS2 is used to simulate and compare the network metrics for both conventional as well as proposed approach and is validated with experimental results over end-to-end delay and message delivery ratio against reported literature.

Q-MOHRA: QoS Assured Multi-Objective Hybrid Routing Algorithm for Heterogeneous WSN

Abstract: Quality of Service (QoS) assurance in Wireless Sensor Network (WSN) is a tough task, and it is more exciting due to the scarcity of resources. The requirement of different WSN applications running over has different constraints. In QoS, routing protocol the network has to balance the traffic. This paper presents a novel heuristic routing algorithm known as QoS assured Multi-objective Hybrid Routing Algorithm (Q-MOHRA) for Heterogeneous WSN. Q-MOHRA takes into account the link (energy, hop count, link quality indicator etc.) and path (jitter) metrics for optimal path selection. The performance of Q-MOHRA is evaluated through intensive simulation and equated with Simple Hybrid Routing Protocol (SHRP) and Dynamic Multi-objective Routing Algorithm (DyMORA). The metrics such as average energy consumption, residual energy, packet delivery ratio, jitter, and normalized routing load are used for comparison. The performance of Q-MOHRA has been observed to outclass SHRP and DyMORA. It improves the packet delivery ratio by 24.31% as compared to SHRP and 11.86% as compared to DyMORA. Q-MOHRA outperforms DyMORA in terms of average energy consumption by a factor of 8.27%.

A deadlock-free routing algorithm for irregular 3D network-on-chips with wireless links

Abstract: In recent years, the idea of wireless three-dimensional network-on-chips (3D NoCs) was promoted in order to design many-core chips with greater performance and lower energy consumption. This technology is the combination of different dies that are stacked on each other. Therefore, it is necessary to propose a suitable routing mechanism for irregular wireless 3D NoCs that can support the agnostic topologies. In this paper, we propose a deadlock-free routing algorithm for wireless 3D NoCs, called Floyd-base Inter-chip Traffic distribution (FIT), which is based on Floyd routing algorithm. In FIT algorithm, the number of hops is reduced compared to the already established deterministic algorithms; moreover, the traffic distribution is improved. Evaluation results show that our proposed routing algorithm significantly improves the performance and throughput by reducing the energy consumption, the average hop count and the communication latency.

Segment Routing interworking with LDP

Abstract: A Segment Routing (SR) node steers a packet through a controlled set of instructions, called segments, by prepending the packet with an SR header. A segment can represent any instruction, topological or service-based. SR allows to enforce a flow through any topological path while maintaining per-flow state only at the ingress node to the SR domain. The Segment Routing architecture can be directly applied to the MPLS data plane with no change in the forwarding plane. This document describes how Segment Routing operates in a network where LDP is deployed and in the case where SR-capable and non-SR-capable nodes coexist.

SDPR: Improving Latency and Bandwidth in On-Chip Interconnect Through Simultaneous Dual-Path Routing

Abstract: Networks-on-chips (NoCs) are gaining in popularity as replacement for shared medium interconnects in chip-multiprocessors (CMPs) and multiprocessor systems-on-chips, and their performance becoming essential to system performance. There have been emerging studies to achieve better power/energy efficiency without performance degradation on NoCs. However, there are still non-negligible latency issues caused by the mechanism of power efficient approaches. To alleviate the latency problem and to transfer data efficiently with the high utilization of interconnect resources, we propose an on-chip network architecture that improves latency and bandwidth. Increasing the data/link widths across the network may considerably resolve this problem but is a costly proposition both in terms of device area and of power. Alternatively, we propose a dual-path router architecture that efficiently exploits path diversity to attain low latency without significant hardware overhead. By: 1) doubling the number of injection and ejection ports; 2) splitting packets into two halves; 3) recomposing routing policy to support path diversity; and 4) provisioning the network hardware design, we can considerably enhance network resource utilization to achieve much higher performance in latency. The proposed simultaneous dual-path routing (SDPR) scheme outperformed the conventional dimension order routing (DOR) technique across synthetic workloads by 31%–40% in average latency and up to a 100% improvement in throughput performance running on a 49-core CMP. Our synthesizable model for the SDPR router and network provides accurate power and area reports. According to the synthesis reports, SDPR incurs insignificant overhead compared to the baseline XY DOR router.

Routing to Multi-Instantiated Destinations: Principles, Practice, and Applications

Abstract: Prior solutions for routing to multi-instantiated destinations simply adapt existing routing algorithms designed for single-instance destinations, or rely on flooding techniques. In this paper, a new approach for routing to multi-instantiated destinations is introduced, and the Multiple Instance Destination Routing (MIDR) framework is presented as an example of the approach. MIDR uses only distance information to multi-instantiated destinations, without routers having to establish overlays, know the network topology, use complete paths to destination instances, or know about all the instances of destinations. MIDR can be used in name-based content routing, IP unicast routing, multicasting, and anycasting; even in scenarios where the network topology is highly dynamic such as in the case of MANETs. It is shown that MIDR provides multiple loop-free paths to destination instances. Extensive simulation-based experiments performed in the context of MANETs show that MIDR outperforms traditional approaches based on unicast protocols and that it scales to large networks.

Energy-Aware Routing Algorithm for Wireless Sensor Networks with Optimal Relay Detecting

Abstract: Efficient and reliable routing plays a key role in wireless sensor networks in which routing design with regard to network availability and node lifetime needs to be deliberately considered. When multi-hop relay transmission is frequently applied to reduce a source node's energy consumption and improve network capacity, a key issue affecting the nodes' participation in the transmission is the problem of suitably determining the next hop in order to prolong each node's lifetime and to maintain the energy-balancing of the whole network. In this study, we propose an energy-aware routing scheme by taking the Cauchy operator, node's residual energy and routing distance into account. Based on Cauchy inequation, we achieve a relationship between the routing distance and the energy usage in the routing. By fixing a relay selection parameter and then identifying the next hop appropriately, we obtain a balancing energy-aware routing algorithm. Numerical results are provided to verify the lifetime and equilibrium of the energy distribution by comparing with them with those of a traditional approach.

Traffic-aware stateless multipath routing for fault-tolerance in IEEE 802.15.4 wireless mesh networks

Abstract: Single-path routing is widely used in wireless networks due to low resource consumption. However, it is vulnerable to link failure because such a failure may adversely affect an entire path. To overcome this, multipath routing has been proposed providing fault-tolerance. In this paper, we propose a novel multipath routing protocol called traffic-aware stateless multipath routing (TSMR) based on an overlaid tree topology comprising two topologies, namely, bounded degree tree (BDT) and root-oriented directional tree (RODT). BDT is strong on reducing routing overhead, and RODT is resilient against lossy links. By synergistically overlaying them, TSMR dynamically selects the local optimal path according to the given traffic flow and the failure on the primary path. In particular, TSMR enables stateless and low overhead routing despite multipath routing by keeping only one-hop neighbors to maintain multiple paths. To evaluate TSMR, we conducted simulations with a shadowing model reflecting lossy links, and compared with single and multipath routing protocols, such as ZTR, STR, AODV, and RPL. The simulation results show that the overall performance of TSMR surpasses that of others for packet delivery ratio, end-to-end delay, control overhead, memory consumption, and power consumption regardless of network size, number of sessions, and traffic flow.

Enhanced reliable reactive routing (ER3) protocol for multimedia applications in 3D wireless sensor networks

Abstract: Sensor networks designed especially for the multimedia applications require high data rate and better Quality of Service (QoS). Offering a reliable and energy efficient routing technique in a harsh and complex three-dimensional (3-D) environment for multimedia applications is a challenging job. Geo-routing and geometric routing have been efficient routing schemes for two-dimensional (2-D), but are unable to work properly for 3-D sensor networks. In order to enhance the resilience to link the dynamics in the 3-D sensor network, in this research an Enhanced Reliable Reactive Routing (ER3) is proposed. ER3 is an advancement to the existing reactive routing schemes, to provide energy efficient and reliable routing of data packets in the complex 3-D sensor networks for multimedia applications. The major attraction of ER3 is its backoff scheme, which occurs in the route discovery phase. In backoff scheme robust pilot paths formed between the source and destination are calculated to enable cooperative forwarding of the data packets. The data packets in ER3 are forwarded greedily to the destination from the source and doesn’t require any prior location information of the nodes. The encompassing simulations suggest that the ER3 outperforms the existing routing protocols on the basis of energy efficiency, low latency and high packet delivery ratio.

Routing Algorithms: Shortest Path, Widest Path, and Spanning Tree

Abstract: In this chapter, we first describe two classes of routing algorithms: shortest path routing and widest path routing. They appear in network routing in many ways and have played critical roles in the development of routing protocols. For example, shortest path routing is used in IP networks is based on the addition cost property. Widest path routing is used for call routing in circuit-switched networks and MPLS networks and is based on the non-additive concave cost property. Widest path routing is also part of quality-of-service (QoS) routing.

Shortest Path Routing Extensions for BGP Protocol

Abstract: Many Massively Scaled Data Centers (MSDCs) have converged on simplified layer 3 routing. Furthermore, requirements for operational simplicity have lead many of these MSDCs to converge on BGP as their single routing protocol for both their fabric routing and their Data Center Interconnect (DCI) routing. This document describes a solution which leverages BGP Link-State distribution and the Shortest Path First algorithm similar to Internal Gateway Protocols (IGPs) such as OSPF.

An Efficient Lookup Search and Forwarding Mechanism for Information-Centric Networking (ICN)

Abstract: Information-centric networking (ICN) is gradually becoming the alternative approach to the traditional Internet architecture through enlightening information (content) distribution on the Internet with content names. The growing rate of Internet traffic has adapted the content-centric architectures to better serve the user requirement of accessing data. Routing disseminates preliminary topology and strategy information as well as long-term changes in them and computes the routing table to coordinate the forwarding procedure. A good routing scheme is essential to reduce delay and to enhance the performance of the network with an efficient searching of the desired content item. In this paper, we present a novel scheme for ICN which evaluates the path for searching the desired content item with an efficient lookup process and we call as Intelligent Routing (IR). Our implement IR scheme utilizes the concept of ternary digital search for making an efficient lookup and evaluates the satisfaction score for making a forwarding decision. The simulation results recommend that IR scheme can reliably accomplish the better performance across other hashing and routing approaches proposed in this field.

All-optical tree-based greedy router using optical logic gates and optical flip-flops

Abstract: Due to ever-increasing throughput demands, the lookup in conventional IP routers based on longest prefix matching is becoming a bottleneck. Additionally, the scalability of current routing protocols is limited by the size of the routing tables. Geometric greedy routing is an alternative to IP routing which replaces longest prefix matching with a simple calculation employing only local information for packet forwarding. For the first time, in this paper we propose a novel and truly all-optical geometric greedy router based on optical logic gates and optical flip-flops. The circuit of the router is constructed through the interconnection of SOAs and directional couplers. The successful functionality of the proposed router is verified through simulation. The circuit enables high data rate throughput.

Algorithms for a special class of state-dependent shortest path problems with an application to the train routing problem

Abstract: We study the state-dependent shortest path problem and focus on its application to the Single Train Routing Problem consisting of a rail network with only double-track segments, where the objective is to route one train through an empty network as fast as possible. We show that the Single Train Routing Problem is NP-hard. We investigate the solution properties and present sufficient conditions for optimality. Different conditions on the parameters are given to guarantee that certain local route selection is optimal. Then, a dynamic programming heuristic is introduced and conditions when the proposed heuristic can obtain the optimal solution in polynomial time are also discussed. Experimental results show the efficiency of the proposed heuristics for general problem settings.