Comparison of formulations and a heuristic for packing Steiner trees in a graph

In this paper, we consider the problem of packing Steiner trees in a graph. This problem arises during the global routing phase of circuit layout design. We consider various integer programming formulations and rank them according to lower bounds they provide as LP-relaxations. We discuss a solution procedure to obtain both lower and upper bounds using one of the LP-relaxations. Computational results to test the effectiveness of our procedures are provided.     

On routing in VLSI design and communication networks

In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. First we propose new and realistic models for both problems. In the global routing problem in VLSI design, we are given a lattice graph and subsets of the vertex set. The goal is to generate trees spanning these vertices in the subsets to minimize a linear combination of overall wirelength (edge length) and the number of bends of trees with respect to edge capacity constraints. In the multicast routing problem in communication networks, a graph is given to represent the network, together with subsets of the vertex set. We are required to find trees to span the given subsets and the overall edge length is minimized with respect to capacity constraints. Both problems are APX-hard. We present the integer linear programming (LP) formulation of both problems and solve the LP relaxations by the fast approximation algorithms for min-max resource-sharing problems in [K. Jansen, H. Zhang, Approximation algorithms for general packing problems and their application to the multicast congestion problem, Math. Programming, to appear, doi:10.1007/s10107-007-0106-8] (which is a generalization of the approximation algorithm proposed by Grigoriadis and Khachiyan [Coordination complexity of parallel price-directive decomposition, Math. Oper. Res. 2 (1996) 321-340]). For the global routing problem, we investigate the particular property of lattice graphs and propose a combinatorial technique to overcome the hardness due to the bend-dependent vertex cost. Finally, we develop asymptotic approximation algorithms for both problems with ratios depending on the best known approximation ratio for the minimum Steiner tree problem. They are the first known theoretical approximation bound results for the problems of minimizing the total costs (including both the edge and the bend costs) while spanning all given subsets of vertices.     

A matheuristic approach for solving a home health care problem

We deal with a Home Health Care Problem (HHCP) which objective consists in constructing the optimal routes and rosters for the health care staffs. The challenge lies in combining aspects of vehicle routing and staff rostering which are two well known hard combinatorial optimization problems. To solve this problem, we initially propose an integer linear programming formulation (ILP) and we tested this model on small instances. To deal with larger instances we develop a matheuristic based on the decomposition of the ILP formulation into two problems. The first one is a set partitioning like problem and it represents the rostering part. The second problem consists in the routing part. This latter is equivalent to a Multi-depot Traveling Salesman Problem with Time Windows (MTSPTW).     

Lower and upper bounds for location-arc routing problems with vehicle capacity constraints

This paper addresses multi-depot location arc routing problems with vehicle capacity constraints. Two mixed integer programming models are presented for single and multi-depot problems. Relaxing these formulations leads to other integer programming models whose solutions provide good lower bounds for the total cost. A powerful insertion heuristic has been developed for solving the underlying capacitated arc routing problem. This heuristic is used together with a novel location–allocation heuristic to solve the problem within a simulated annealing framework. Extensive computational results demonstrate that the proposed algorithm can find high quality solutions. We also show that the potential cost saving resulting from adding location decisions to the capacitated arc routing problem is significant.     

A cyclic scheduling problem with an undetermined number of parallel identical processors

This paper presents two integer linear programming (ILP) models for cyclic scheduling of tasks with unit/general processing time. Our work is motivated by digital signal processing (DSP) applications on FPGAs (Field-Programmable Gate Arrays)—hardware architectures hosting several sets of identical arithmetic units. These hardware units can be formalized as dedicated sets of parallel identical processors. We propose a method to find an optimal periodic schedule of DSP algorithms on such architectures where the number of available arithmetic units must be determined by the scheduling algorithm with respect to the capacity of the FPGA circuit. The emphasis is put on the efficiency of the ILP models. We show the advantages of our models in comparison with common ILP models on benchmarks and randomly generated instances.     

Performance-Driven Layer Assignment by Integer Linear Programming and Path-Constrained Hypergraph Partitioning

Performance-driven physical layout design is becoming increasingly important for both high speed integrated circuits and printed circuit boards. This paper studies the problem of assigning wire segments into two layers so as to minimize the number of vias, while taking into account performance constraints such as layer preference and circuit timing. We show that using the Elmore delay model, three timing problems in synchronous digital circuits—the long path problem, the short path problem and the time skew problem—can be formulated as a set of linear inequalities. We use the model of signed hypergraph to represent two-layer routings and formulate the performance-driven optimum layer assignment problem as the path-constrained maximum balance problem in a signed hypergraph. Two solution methods are developed and implemented. First, an integer linear programming formulation is derived for finding exact solutions. Second, a local-search heuristic for hypergraph partitioning is extended to cope with path-inequality constraints. Experimental results on a set of layer-assignment benchmarks demonstrated that the path-constrained local-search heuristic achieves optimum or near-optimum solutions with several orders of magnitude faster than the integer linear programming approach.     

The offshore wind farm array cable layout problem: a planar open vehicle routing problem

In an offshore wind farm (OWF), the turbines are connected to a transformer by cable routes that cannot cross each other. Finding the minimum cost array cable layout thus amounts to a vehicle routing problem with the additional constraints that the routes must be embedded in the plane. For this problem, both exact and heuristic methods are of interest. We optimize cable layouts for real-world OWFs by a hop-indexed integer programming formulation, and develop a heuristic for computing layouts based on the Clarke and Wright savings heuristic for vehicle routing. Our heuristic computes layouts on average only 2% more expensive than the optimal layout. Finally, we present two problem extensions arising from real-world OWF cable layouts, and adapt the integer programming formulation to one of them. The thus obtained optimal layouts are up to 13% cheaper than the actually installed layouts.     

Some integer programs arising in the design of main frame computers

In this paper we describe and discuss a problem that arises in the (global) design of a main frame computer. The task is to assign certain functional units to a given number of so called multi chip modules or printed circuit boards taking into account many technical constraints and minimizing a complex objective function. We describe the real world problem. A thorough mathematical modelling of all aspects of this problem results in a rather complicated integer program that seems to be hopelessly difficult — at least for the present state of integer programming technology. We introduce several relaxations of the general model, which are alsoNP-hard, but seem to be more easily accessible. The mathematical relations between the relaxations and the exact formulation of the problem are discussed as well.On leave from University of São Paulo, Brazil.     

A hybrid constraint programming approach to?the?log-truck scheduling problem

Scheduling problems in the forest industry have received significant attention in the recent years and have contributed many challenging applications for optimization technologies. This paper proposes a solution method based on constraint programming and mathematical programming for a log-truck scheduling problem. The problem consists of scheduling the transportation of logs between forest areas and woodmills, as well as routing the fleet of vehicles to satisfy these transportation requests. The objective is to minimize the total cost of the non-productive activities such as the waiting time of trucks and forest log-loaders and the empty driven distance of vehicles. We propose a constraint programming model to address the combined scheduling and routing problem and an integer programming model to deal with the optimization of deadheads. Both of these models are combined through the exchange of global constraints. Finally the whole approach is validated on real industrial data.     

An algorithm and upper bounds for the weighted maximal planar graph problem

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.     

The wafer probing scheduling problem (WPSP)

The wafer probing scheduling problem (WPSP) is a variation of the parallel-machine scheduling problem, which has many real-world applications, particularly, in the integrated circuit (IC) manufacturing industry. In the wafer probing factories, the jobs are clustered by their product types, which must be processed on groups of identical parallel machines and be completed before the due dates. Further, the job processing time depends on the product type, and the machine setup time is sequence dependent on the orders of jobs processed. Since the wafer probing scheduling problem involves constraints on job clusters, job-cluster dependent processing time, due dates, machine capacity, and sequence dependent setup time, it is more difficult to solve than the classical parallel-machine scheduling problem. In this paper, we formulate the WPSP as an integer programming problem. We also transform the WPSP into the vehicle routing problem with time windows (VRPTW), a well-known network routing problem which has been investigated extensively. An illustrative example is given to demonstrate the proposed transformation. Based on the provided transformation, we present three efficient algorithms to solve the WPSP near-optimally.     

Operational aircraft maintenance routing problem with remaining time consideration

The aircraft maintenance routing problem is one of the most studied problems in the airline industry. Most of the studies focus on finding a unique rotation that will be repeated by each aircraft in the fleet with a certain lag. In practice, using a single rotation for the entire fleet is not applicable due to stochasticity and operational considerations in the airline industry. In this study, our aim is to develop a fast responsive methodology which provides maintenance feasible routes for each aircraft in the fleet over a weekly planning horizon with the objective of maximizing utilization of the total remaining flying time of fleet. For this purpose, we formulate an integer linear programming (ILP) model by modifying the connection network representation. The proposed model is solved by using branch-and-bound under different priority settings for variables to branch on. A heuristic method based on compressed annealing is applied to the same problem and a comparison of exact and heuristic methods are provided. The model and the heuristic method are extended to incorporate maintenance capacity constraints. Additionally, a rolling horizon based procedure is proposed to update the existing routes when some of the maintenance decisions are already fixed.     

Polyhedral results for the edge capacity polytope

Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.?As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated.?The derived inequalities are tested on (i) the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems. Received: September 2000 / Accepted: October 2001?Published online March 27, 2002     

Finding multiple solutions to general integer linear programs

Integer linear programming (ILP) problems occur frequently in many applications. In practice, alternative optima are useful since they allow the decision maker to choose from multiple solutions without experiencing any deterioration in the objective function. This study proposes a general integer cut to exclude the previous solution and presents an algorithm to identify all alternative optimal solutions of an ILP problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method.     

Integer linear programming models for a cement delivery problem

We consider a cement delivery problem with an heterogeneous fleet of vehicles and several depots. The demands of the customers are typically larger than the capacity of the vehicles which means that most customers are visited several times. This is a split delivery vehicle routing problem with additional constraints. We first propose a two phase solution method that assigns deliveries to the vehicles, and then builds vehicle routes. Both subproblems are formulated as integer linear programming problems. We then show how to combine the two phases in a single integer linear program. Experiments on real life instances are performed to compare the performance of the two solution methods.     

An integer linear formulation for the file transfer scheduling problem

In this paper, we propose a new integer linear programming (ILP) formulation for solving a file transfer scheduling problem (FTSP), which is to minimize the overall time needed to transfer all files to their destinations for a given collection of various sized files in a computer network. Each computer in this network has a limited number of communication ports. The described problem is proved to be NP-hard in a general case. Our formulation enables solving the problem by standard ILP solvers like CPLEX or Gurobi. To prove the validity of the proposed ILP formulation, two new reformulations of FTSP are presented. The results obtained by CPLEX and Gurobi solvers, based on this formulation, are presented and discussed.     

SCIP: solving constraint integer programs

Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and non-commercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly non-linear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current state-of-the-art techniques for proving the validity of properties on circuits containing arithmetic.      

Confidence intervals in the solution of stochastic integer linear programming problems

A method is proposed to estimate confidence intervals for the solution of integer linear programming (ILP) problems where the technological coefficients matrix and the resource vector are made up of random variables whose distribution laws are unknown and only a sample of their values is available. This method, based on the theory of order statistics, only requires knowledge of the solution of the relaxed integer linear programming (RILP) problems which correspond to the sampled random parameters. The confidence intervals obtained in this way have proved to be more accurate than those estimated by the current methods which use the integer solutions of the sampled ILP problems.This research was partially supported by the Italian National Research Council contract no. 82.001 14.93 (P.F. Trasporti).     

A filled function method for finding a global minimizer on global integer optimization

The paper gives a definition of the filled function for nonlinear integer programming. This definition is modified from that of the global convexized filled function for continuous global optimization. A filled function with only one parameter which satisfies this definition is presented. We also discuss the properties of the proposed function and give a filled function method to solve the nonlinear integer programming problem. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.     

AN EFFECTIVE DETAILED ROUTING ALGORITHM CONSIDERING ADVANCED TECHNOLOGY NODES

Detailed routing has become much challenging in modern circuit designs due to the extreme scaling of chip size and the complicated design rules. In this paper, we give an effective algorithm for detailed routing considering advanced technology nodes. First, we present a valid pin-access candidates generation technology for handling complex pin shapes. Then, we propose a tree-based nets components selection algorithm to decide connecting order for multiple nets components. Finally, combined with global routing results and advanced technology nodes, an initial routing results optimization algorithm is presented to achieve the final detailed routing results. Experimental results on industry benchmarks show that, our proposed algorithm not only achieves 100%routability on real industrial cases in a reasonable runtime, but also optimizes total wirelength, total vias and other advanced technology nodes simultaneously.