An ideal column algorithm for integer programs with special ordered sets of variables

This paper addresses the close-enough traveling salesman problem, a variant of the Euclidean traveling salesman problem, in which the traveler visits a node if it passes through the neighborhood set of that node. We apply an effective strategy to discretize the neighborhoods of the nodes and the carousel greedy algorithm to appropriately select the neighborhoods that, step by step, are added to the partial solution until a feasible solution is generated. Our heuristic, based on these ingredients, is able to compute tight upper and lower bounds on the optimal solution relatively quickly. The computational results, carried out on benchmark instances, show that our heuristic often finds the optimal solution, on the instances where it is known, and in general, the upper bounds are more accurate than those from other algorithms available in the literature. Summary of Contribution: In this paper, we focus on the close-enough traveling salesman problem. This is a problem that has attracted research attention over the last 10 years; it has numerous real-world applications. For instance, consider the task of meter reading for utility companies. Homes and businesses have meters that measure the usage of gas, water, and electricity. Each meter transmits signals that can be read by a meter reader vehicle via radio-frequency identification (RFID) technology if the distance between the meter and the reader is less than r units. Each meter plays the role of a target point and the neighborhood is a disc of radius r centered at each target point. Now, suppose the meter reader vehicle is a drone and the goal is to visit each disc while minimizing the amount of energy expended by the drone. To solve this problem, we develop a metaheuristic approach, called (lb/ub)Alg, which computes both upper and lower bounds on the optimal solution value. This metaheuristic uses an innovative discretization scheme and the Carousel Greedy algorithm to obtain high-quality solutions. On benchmark instances where the optimal solution is known, (lb/ub)Alg obtains this solution 83% of the time. Over the remaining 17% of these instances, the deviation from the optimality is 0.05%, on average. On the instances with the highest overlap ratio, (lb/ub)Alg does especially well.

The weighted maximal planar graph (WMPG) is practically important in the laying out of facilities in modern manufacturing environments. Given a weighted complete graph, the WMPG seeks to find a sub-graph such that it is planar—it can be embedded on the plane without any arcs intersecting, and it is maximal—no additional arc can be added to the sub-graph without destroying its planarity, and it also has the maximal sum of arc weights. In this paper, an integer linear programming (ILP) model is newly introduced for the problem. Two meta-heuristics are then derived from the ILP relaxation. The first meta-heuristic considers all variables with fractional values greater than half in the ILP relaxation to build an initial sub-graph from which a planar sub-graph is extracted using greedy random adaptive search procedure (GRASP) and augmented by triangulation of faces. The second meta-heuristic considers only arcs with integer values in the ILP relaxation. The remaining arcs are then sorted in descending order of their weights, for selection and insertion with a planarity testing procedure, to obtain a feasible solution using GRASP. Computational results are reported on a set of 100 test instances of size varying from 20 to 100 facilities. The computational results demonstrate the tightness of the new upper bound when compared to the classical one as well as the good performance of the proposed metaheuristics when compared to the best-known procedures in the literature in terms of solution quality and computational requirement. Finally, the paper presents a successful integration of GRASP with classical optimisation approaches and should be attempted for other optimisation problems.

Integer Linear Programming (ILP) is commonly used in high level and system level synthesis. It is an NP-Complete problem (in general cases). There exists some tools that give an optimal solution for small ILP formulations. Nevertheless, these tools may not give solutions for complex formulations. In this paper, we present a solution to overcome the problem of complexity in ILP formulations. We propose a partitioning methodology based on a constraint graph representing all the constraints included in any ILP formulation. To direct the partitioning, the constraint graph nodes are grouped to represent Data Flow Graph (DFG) nodes. This reduced constraint graph can be used to partition any ILP formulation based on DFG. We illustrate this method on ILP formulation for scheduling. Results on ILP scheduling formulations are promising.

Most integer programming algorithms involve at least some binary data handling, e.g., set membership, intersection, and union. This paper presents some of the important concerns involved in implementation. When the algorithms are coded for the computer an effort is sometimes made to take advantage of the basic binary configuration of the digital computer to aid in the storage and handling of such data. In the past this has meant the use of assembly language subroutines. This paper demonstrates a procedure for achieving the same objectives in the main program using FORTRAN. The obvious advantage of the procedure is the elimination of the several storage and retrieval operations for register values as the program shifts control from the main routine to the subroutine and back again. Of particular interest are (1) the use of the computer arithmetic to achieve the relative complement operation and (2) the importance of additional savings due to overhead reduction when bit storage is used. Computational results are presented comparing various regimes for storing and handling binary data. In addition, an analytic procedure for comparing the storage requirements and execution times of different data structures is demonstrated for a particular integer programming application. This paper should serve as a useful tutorial for anyone developing and/or implementing integer programming algorithms.

Different integer linear programming (ILP) formulations have been proposed for the routing and wavelength assignment problem in WDM optical networks, mainly for asymmetrical systems, more than for symmetrical systems, under different objectives. We propose a synthesis of the mathematical models for symmetrical systems, with a unified and simplified notation for four widely used objectives. As for asymmetrical traffic models (Jaumard, Meyer, and Thiongane, 2004), we show that all formulations, both link and path formulations, are equivalent in terms of the upper bound value provided by the optimal solution of their linear programming relaxation, although their number of variables and constraints differ. We propose an experimental comparison of the best linear relaxation bounds with the optimal ILP solutions whenever it is possible, for several network and traffic instances.

The code quality of many high-level language compilers in the field of digital signal processing is not satisfactory. This is mostly due to the complexity of the code generation problem together with the irregularity of typical DSP architectures. Since digital signal processors mostly are traded on the high volume consumer market, they are subject to serious cost constraints. On the other hand, many embedded applications demand high performance capacities. Thus, it is very important that the features of the processor are exploited as efficiently as possible. By using integer linear programming (ILP), the deficiencies of the decoupling of different code generation phases can be removed, since it is possible to integrate instruction scheduling and register assignment in one homogeneous problem description. This way, optimal solutions can be found-albeit at the cost of high compilation times. Our experiments show, that approximations based on integer linear programming can provide a better solution quality than classical code generation algorithms in acceptable runtime for medium sized code sequences. The experiments were performed for a modern DSP, the Analog Devices ADSP-2106x.

This paper considers the problem of communication coverage for sustainable data forwarding in wireless sensor networks, where an energy-aware deployment model of relay nodes (RNs) is proposed. The model used in this paper considers constrained placement and is different from the existing one-tiered and two-tiered models. It supposes two different types of sensor nodes to be deployed, i) energy rich nodes (ERNs), and ii) energy limited nodes (ELNs). The aim is thus to use only the ERNs for relaying packets, while ELN's use will be limited to sensing and transmitting their own readings. A minimum number of RNs is added if necessary to help ELNs. This intuitively ensures sustainable coverage and prolongs the network lifetime. The problem is reduced to the traditional problem of minimum weighted connected dominating set (MWCDS) in a vertex weighted graph. It is then solved by taking advantage of the simple form of the weight function, both when deriving exact and approximate solutions. Optimal solution is derived using integer linear programming (ILP), and a heuristic is given for the approximate solution. Upper bounds for the approximation of the heuristic (versus the optimal solution) and for its runtime are formally derived. The proposed model and solutions are also evaluated by simulation. The proposed model is compared with the one-tiered and two-tiered models when using similar solution to determine RNs positions, i.e., minimum connected dominating set (MCDS) calculation. Results demonstrate the proposed model considerably improves the network life time compared to the one-tiered model, and this by adding a lower number of RNs compared to the two-tiered model. Further, both the heuristic and the ILP for the MWCDS are evaluated and compared with a state-of-the-art algorithm. The results show the proposed heuristic has runtime close to the ILP while clearly reducing the runtime compared to both ILP and existing heuristics. The results also demonstrate scalability of the proposed solution.

Multi-Robot Path Planning for Each Robot with Several Jobs in a Single Trip

The paper presents the Integer Programming (IP) algorithms in mining a compact rough classification model. The algorithm is based on creating a 0--1 integer programming model from a rough discernibility relations of a decision system (DS) to find minimum selection of important attributes which is called reduct in rough set theory. A branch and bound search strategy that performs a non-chronological backtracking is proposed to solve the problem. The experimental result shows that the proposed IP algorithm has significantly reduced the number of rules generated from the obtained reducts with high percentage of classification accuracy. The branch and bound search strategy has shown reduction in certain amount of search.

Exact solution approaches for integer linear generalized maximum multiplicative programs through the lens of multi-objective optimization

Tabu search techniques have been applied to a wide class of optimization problems. Tabu search is a meta procedure that requires a subroutine to find local optima. In most examples of tabu search the subroutine used is problem specific. However, in order to apply tabu search successfully to general 0-1 integer programs, the subroutine must be general. The pivot and complement heuristic is a procedure that frequently finds feasible solutions for general 0-1 integer programs. In this paper we implement several elements of strategies proposed by tabu search method that result in general heuristics for 0-1 integer programs that could be included in an integer programming code. These heuristics employ the ZOOM/XMP implementation of the pivot and complement heuristic as a subroutine. Computational results indicate that these heuristics provide good bounds for many problems. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

We discuss the convergence of cutting plane algorithms for a class of nonconvex programs called the Generalized Lattice Point Problems (GLPP). A set of sufficient conditions which guarantee finite convergence are presented. Although these conditions are usually difficult to enforce in a practical implementation, they do illustrate the various factors that must be involved in a convergent rudimentary cutting plane algorithm. A striking example of nonconvergence (in which no subsequence converges to a feasible solution, even when seemingly strong cutting planes are used), is presented to show the effect of neglecting one such factor. We give an application of our analysis to problems with multiple choice constraints and finally discuss a modification of cutting plane algorithms so as to make finite convergence more readily implementable.

Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation to the striking observation that approximations based on linear programming (LP) relaxations are often tight on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that tightness generalizes from train to test data.

This paper describes the details of a successful application where an integer programming and evolutionary hybrid algorithm was used to solve a bus driver duty optimization problem. The task is NP-hard, therefore theoretically optimal solutions can only be calculated for very small problem instances. Our aim is to obtain solutions of good quality within reasonable time limits. We first applied an integer programming approach to a set partitioning problem. The model was solved with a column generation algorithm in a branch and bound scheme. In order to solve larger real-life problems, we have combined the integer programming method with a greedy 1+1 steady state evolutionary algorithm. The resulting hybrid algorithm was capable of providing near-optimal solutions within reasonable timescales to larger instances of the bus driver scheduling problem. We present the results and running times of our algorithm in detail, as well as possible directions of future improvements.

Discrete search path planning is known to be a NP-hard problem, and problem-solving methods proposed so far rely on heuristics with no way to reasonably estimate solution quality for practical size problems. Departing from traditional nonlinear model formulations, a novel information theoretic based approach using integer linear programming (ILP) is proposed to optimally solve the discrete open-loop centralized search path planning problem with anticipated feedback, involving a team of homogeneous agents. The approach takes advantage of objective function separability and conditional probability independence of observations to efficiently minimize expected system entropy. A network representation is exploited to simplify modeling, reduce constraint specification and speed-up problem-solving. The novelty of the approach consists in capturing false-alarm observations explicitly while proposing an original and efficient way to linearize the underlying non-linear expected entropy function required to properly represent target location uncertainty, making for the first time practical problems tractable. The proposed ILP formulation rapidly yields near-optimal solutions for realistic problems while providing for the first time, a robust lower bound through Lagrangian relaxation. Long planning problem horizon may be dynamically adapted by periodically solving new problem instances incorporating actual observation outcomes from previous episodes over receding horizons. Computational results clearly show the value of the approach in comparison to a myopic heuristic over various problem instances.