Node Packing Research Articles

Leafy Crossed Cube: An Efficient PIN for High Speed Computing

This paper attempts to suggest a new derived parallel interconnection network for high end computing system namely the Leafy Cross cube (LCC). As compared to the other interconnection networks LCC is found to be more attractive in terms of topological parameters such as diameter, average node distance and message traffic density etc. It helps to improving the node packing density in the interconnection network.

Reliable, Effective and Fault-Tolerant Design of LeafyCube Interconnection Network Topology

This paper attempts to derive the performance properties of the Leafycube (LC) interconnection network. The Leafycube is already observed to have quite superior topological properties in comparison to the other contemporary networks. The various performance parameters of the LC network are studied and compared with the existing HC and its variants. The routing and broadcasting algorithms are proposed and the time complexities are also compared. The paper attempts to evaluate the cost effectiveness, reliability and fault tolerance aspects of LC interconnection network in order to justify the novelty in the design of the proposed structure. The leafy structure helps to retain the original hypercube while improving the node packing density in the interconnection network.

Effective methods for solving the Bi-criteria p-Center and p-Dispersion problem

Given a collection of n locations and a symmetric measure of distance (difference) between each pair of locations, we seek to identify (select) a subset of p locations so as to achieve two distinct objectives. The first objective is to minimize the maximum dissimilarity (i.e., distance) between the selected locations and other locations. The second objective is to maximize the minimum distance (diversity) among the selected locations themselves. Based on the relationship between the max-min diversity problem and the node packing problem, we propose an integer programming (IP) model and a corresponding incremental algorithm to find the non-dominated frontier for this bi-criteria optimization problem. Subsequently we use the relationship between this IP model and a corresponding set-covering problem to propose effective methods for solving the former. Finally we employ the relationship between the node packing constraints and the set covering constraints to propose a new family of valid inequalities for the corresponding IP models that are potentially effective when solving large instances of these models. Through computational experiments we demonstrate the effectiveness of our proposed methods for solving relatively large instances of this bi-criteria optimization problem.

An integer programming approach for solving the p-dispersion problem

Given a collection of n items (elements) and an associated symmetric distance dij between each pair of items i and j, we seek a subset P of these items (with a given cardinality p) so that the minimum pairwise distance among the selected items is maximized. This problem is known as the max–min diversity problem or the p-dispersion problem, and it is shown to be np-hard. We define a collection of node packing problems associated with each instance of this problem and employ a binary search among these node packing problems to devise an effective procedure for solving the original problem. We employ existing integer programming techniques, i.e., branch-and-bound and strong valid inequalities, to solve these node packing problems. Through a computational experiment we show that this approach can be used to solve relatively large instances of the p-dispersion problem, i.e., instances with more than 1000 items. We also discuss an application of this problem in the context of locating traffic sensors in a highway network.

Strong formulation for the spot 5 daily photograph scheduling problem

Earth observation satellites, such as the SPOT 5, take photographs of the earth according to consumers' demands. Obtaining a good schedule for the photographs is a combinatorial optimization problem known in the literature as the daily photograph scheduling problem (DPSP). The DPSP consists of selecting a subset of photographs, from a set of candidates, to different cameras, maximizing a profit function and satisfying a large number of constraints. Commercial solvers, with standard integer programming formulations, are not able to solve some DPSP real instances available in the literature. In this paper we present a strengthened formulation for the DPSP, based on valid inequalities arising in node packing and 3-regular independence system polyhedra. This formulation was able, with a commercial solver, to solve to optimality all those instances in a short computation time.

Node packings on cocomparability graphs

We give a compact formulation for the clique inequalities defining the fractional node packing polytope on cocomparability graphs.

The relation of time indexed formulations of single machine scheduling problems to the node packing problem

The relation of time indexed formulations of nonpreemptive single machine scheduling problems to the node packing problem is established and then used to provide simple and intuitive alternate proofs of validity and maximality for previously known results on the facial structure of the scheduling problem. Previous work on the facial structure has focused on describing the convex hull of the set of feasible partial schedules, schedules in which not all jobs have to be started. The equivalence between the characteristic vectors of this set and those of the set of feasible node packings in a graph whose structure is determined by the parameters of the scheduling problem is established. The main contribution of this paper is to show that the facet inducing inequalities for the convex hull of the set of feasible partial schedules that have integral coefficients and right hand side 1 or 2 are the maximal clique inequalities and the maximally and sequentially lifted 5-hole inequalities of the convex hull of the set of feasible node packings in this graph respectively.

Transitive Packing: A Unifying Concept in Combinatorial Optimization

This paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope include the node packing, acyclic subdigraph, bipartite subgraph, planar subgraph, clique partitioning, partition, transitive acyclic subdigraph, interval order, and relatively transitive subgraph polytopes. We give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope, thereby introducing generalized cycle, generalized clique, generalized antihole, generalized antiweb, and odd partition inequalities. On the one hand, these classes subsume several known classes of valid inequalities for several special cases; on the other hand, they yield many new inequalities for several other special cases. For some of the classes we also prove a lower bound on their Gomory--Chvatal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering, as well as to balanced and ideal matrices.

Routing trains through a railway station based on a node packing model

In this paper, we describe the problem of routing trains through a railway station. This routing problem is a subproblem of the automatic generation of timetables for the Dutch railway system. The problem of routing trains through a railway station is the problem of assigning each of the involved trains to a route through the railway station, given the detailed layout of the railway network within the station and given the arrival and departure times of the trains. When solving this routing problem, several aspects such as capacity, safety, and customer service have to be taken into account. In this paper, we describe this routing problem in terms of a weighted node packing problem. Furthermore, we describe an algorithm for solving this routing problem to optimality. The algorithm is based on preprocessing, valid inequalities, and a branch-and-cut approach. The preprocessing techniques aim at identifying superfluous nodes which can be removed from the problem instance. The characteristics of the preprocessing techniques with respect to propagation are investigated. We also present the results of a computational study in which the model, the preprocessing techniques and the algorithm are tested based on data related to the railway stations Arnhem, Hoorn and Utrecht CS in the Netherlands.

Valid inequalities for a class of assembly system problems

This paper establishes a foundation for a cutting plane approach to solve a class of problems that are important to assembly systems, including assembly line balancing (ALB), workload smoothing (WS) and assembly system design (ASD). We describe relationships among the integer programming formulations of these problems and recast their relaxations as node packing problems (NPPs). Exploiting the well-known structure of the NPP and other relationships among the assembly problems, we derive several families of valid inequalities and show conditions under which they also define facets for the associated ALB, WS and ASD polytopes.

Facets for node packing

The Node Packing Problem is an extremely important problem given that it comprises the underlying structure of numerous optimization problems either directly or indirectly. This paper presents a constraint approach which produces new facets for the Node Packing Problem. A number of different problem applications are solved incorporating this new constraint approach using a commercial software package on a personal computer demonstrating the effectiveness of the underlying facets in practice. The new facet structures provide a means for addressing general dispersion and separation requirements using mathematical programming.

Maximizing a submodular function by integer programming: Polyhedral results for the quadratic case

We give a new mixed integer programming (MIP) formulation for the quadratic cost partition problem that is derived from a MIP formulation for maximizing a submodular function. Several classes of valid inequalities for the convex hull of the feasible solutions are derived using the valid inequalities for the node packing polyhedron. Facet defining conditions and separation algorithms are discussed and computational results are reported.

Node distribution and packing density in the rat CNS‐PNS transitional zone

Node distribution and packing density in the rat CNS‐PNS transitional zone

Node distribution and packing density in the rat CNS-PNS transitional zone.

The density of nodes of Ranvier was examined at CNS, PNS, and transitional zone (TZ) levels of rat lumbar ventral motoneurone fibres. It was found to be significantly greater in the TZ than at the other levels: The difference was sevenfold for the ventral root and at least fourfold for central fibre levels. Node distribution and spacing was examined within the two main types of TZ found in rat ventral rootlets: the first, in which the TZ is short and is approximately on a level with the surface of the cord; and the second, in which it is much longer and extends into the proximal part of the rootlet. Node spacing was estimated as nearest neighbour distance, the true distance between adjacent node centres. This is a better estimate of node spacing than simple density since it measures the actual linear distance between nodes over which any interaction between them would be likely to take place. Despite marked differences in the dimensions of the two types of TZ, nearest neighbour distance distribution was very similar in each, suggesting that similar mechanisms may influence their spacing during development. The TZ contains especially large amounts of interstitial tissue, mainly composed of astrocyte processes, separating the fibres traversing it. The proportion of the TZ composed of interstitium was over three times that in the ventral root and nearly twice that at the CNS level studied. The large amounts of astrocytic tissue in the TZ may be related to the high packing density of nodes. It may function to regulate extracellular ionic concentrations in the TZ and to maintain a stable ionic environment for the transitional nodes.

Routing Trains Through Railway Stations: Model Formulation and Algorithms

In this paper we consider the problem of routing trains through railway stations. This problem occurs as a subproblem in a project which the authors are carrying out in cooperation with the Dutch railways. The project involves the analysis of future infrastructural capacity requirements in the Dutch railway network. Part of this project is the automatic generation and evaluation of timetables. To generate a timetable a hierarchical approach is followed: at the upper level in the hierarchy a tentative timetable is generated, taking into account the specific scheduling problems of the trains at the railway stations at an aggregate level. At the lower level in the hierarchy it is checked whether the tentative timetable is feasible with respect to the safety rules and the connection requirements at the stations. To carry out this consistency check, detailed schedules for the trains at the railway yards have to be generated. In this paper we present a mathematical model formulation for this detailed scheduling problem, based on the Node Packing Problem (NPP). Furthermore, we describe a solution procedure for the problem, based on a branch-and-cut approach. The approach is tested in an empirical study with data from the station of Zwolle in The Netherlands.

Symplectic finite element scheme: application to a driven problem with a regular singularity

A new finite element (FE) scheme, based on the decomposition of a second order differential equation into a set of first order symplectic (Hamiltonian) equations, is presented and tested on a one-dimensional, driven SturmdashLiouville problem. Error analysis shows improved cubic convergence in the energy norm for piecewise linear “tent” elements, as compared to quadratic convergence for the standard and symplectic hybrid (i.e. ‘tent’ and piecewise constant) FE methods. The convergence deteriorates in the presence of a regular singular point, but can be recovered by appropriate mesh node packing. Optimal mesh packing exponents are derived to ensure cubic (respectively quadratic for the hybrid FE method) convergence with minimal numerical error. The symplectic hybrid FE scheme is shown to be approximately 30–40 times more accurate than the standard FE scheme, for an exact test problem based on determining the nonideal magnetohydrodynamic stability of a fusion plasma. A further suppression of the error by one order of magnitude is achieved for the symplectic tent element method.

A Strong Cutting Plane/Branch-and-Bound Algorithm for Node Packing

We present a strong cutting plane/branch-and-bound algorithm for node packing. The cutting planes are obtained from cliques and lifting odd hole inequalities. Computational results are reported.

A Strong Cutting Plane/Branch-and-Bound Algorithm for Node Packing

We present a strong cutting plane/branch-and-bound algorithm for node packing. The cutting planes are obtained from cliques and lifting odd hole inequalities. Computational results are reported.

Random near-regular graphs and the node packing problem

Nemhauser and Trotter [12] proposed a certain easily-solved linear program as a relaxation of the node packing problem. They showed that any variables receiving integer values in an optimal solution to this linear program also take on the same values in an optimal solution to the (integer) node packing problem. Let π be the property of graphs defined as follows: a graph G has property π if and only if there is a unique optimal solution to the linear-relaxation problem, and this solution is completely fractional. If a graph G has property π then no information about the node packing problem on G is gained by solving the linear relaxation. We calculate the asymptotic probability that a certain type of ‘sparse’ random graph has property π, as the number of its nodes tends to infinity. Let m be a fixed positive integer, and consider the following random graph on the node set {1,2 …, n}). We join each node, j say, to exactly m other nodes chosen randomly with replacement, by edges oriented away from j; we denote by G n ( m) the undirected graph obtained by deleting all orientations and allowing all parallel edges to coalesce. We show that, as n → ∞, P(G n(m) has property π)→ 0 if m = 1, 1 if m ⩾ 3, and we conjecture that P(G n(2) has property π)→ (1–2e −2) 1 2 .

A note on node packing polytopes on hypergraphs

A facial structure of the node packing polytope, i.e., of the convex hull of integer solutions of the node packing problem, on hypergraphs is studied. First, the results extracted by Chvàtal and by Balas and Zemel on canonical facets of the node packing polytopes on graphs are generalized to derive some specific hypergraphs having canonical facets. Second, it is shown that the facial structure of the node packing polytope on a hypergraph, named a fat graph, has a very close relationship to the facial structures of the node packing polytope and also of the convex hull of integer solutions of an integer linear program, which are defined on a specific graph corresponding to the fat graph.