ILP Solver Research Articles

The Selective Fixing Algorithm for the closest string problem

A hybrid heuristic algorithm based on integer linear programming is proposed for the closest string problem (CSP). The algorithm takes a rough feasible solution in input and iteratively selects variables to be fixed at their initial value until the number of free variables is small enough for the remaining problem to be solved to optimality by an ILP solver. The new solution can then be used as input for another iteration of the algorithm and this approach is repeated a predefined number of times. The procedure is denoted as Selective Fixing Algorithm (SFA). SFA has first been tested on standard instances available from the literature, which is denoted as rectangular having string length larger than the number of strings. Then, this approach has also been tested on the so-called square instances (having string length equal to the number of strings) and rectangular inverse instances (having string length smaller than the number of strings). Computational experiments indicate that SFA globally outperforms the state-of-the-art heuristics.

MBA Student Sectioning

Maastricht University is offering a MBA program for people that have a bachelor degree and at least 5 years of working experience. Within the MBA program, students work in groups of 5 during a two year cycle. This thesis is about the formation of the student groups. The MBA program contains 60 students. Every year, two intake moments take place that usually allow 15 new students to enter. All 60 students follow the same course at the same time, implying that the order in which a student follows the courses depends only on the moment at which he/she starts the program. Every two periods, The university creates new student groups according to a set of hard and soft constraints, such that well-diversified groups are formed. Therefore, the student-with-student history, gender, nationality, and level of expertise of each student is taken into account. Hence a mapping from a set of students to groups is created that takes into account the corresponding constraints. The university chooses a group leader for each group. Two general solution methods are applied to the MBA sectioning problem. The first method uses the simplex algorithm to solve the problem. Therefore an integer linear program formulation of the problem was needed, and used as an input for an efficient ILP solver. The second approach starts with an initial feasible solution and improves upon this feasible solution using different improvement algorithms. The quality of each feasible solution depends on the calculated objective function value that measures the level of satisfaction of the different constraints. Different initial solution and improvement algorithms are discussed that help to obtain a feasible solution with an objective function value that is as low as possible. The implemented improvement algorithms are the Descent Improvement algorithm, Tabu Search, Simulated Annealing, and the Bipartite Weighted Matching Improvement algorithm. The first three algorithms make individual students swap between existing group formations. The Bipartite Weighted Matching Improvement algorithm iteratively selects a student from each group, and finds local optimal solutions for a bipartite matching problem in order to improve the overall objective value of the whole problem. In order to test the algorithms, one has to make sure that the instance on which the algorithms are tested mimics a real life example. Therefore, a simulation program is established that mimics the two year cycle and produces such an instance. Empirical results show that the best improvement algorithm considered is the Bipartite Weighted Matching Improvement algorithm. This algorithm, combined with an initial solution algorithm, is now being implemented into the current computer system of Maastricht University.

Solving the Crop Allocation Problem using Hard and Soft Constraints

Application tools for the crop allocation problem (CAP) are required for agricultural advisors to design more efficient farming systems. Despite the extensive treatment of this issue by agronomists in the past, few methods tackle the crop allocation problem considering both the spatial and the temporal aspects of the CAP. In this paper, we precisely propose an original formulation addressing the crop allocation planning problem while taking farmers' management choices into account. These choices are naturally represented by hard and soft constraints in the Weighted CSP formalism. We illustrate our proposition by solving a medium-size virtual farm using either a WCSP solver (toulbar2) or an ILP solver (NumberJack/SCIP). This preliminary work foreshadows the development of a decision-aid tool for supporting farmers in their crop allocation strategies.

Cutting plane versus compact formulations for uncertain (integer) linear programs

Robustness is about reducing the feasible set of a given nominal optimization problem by cutting “risky” solutions away. To this end, the most popular approach in the literature is to extend the nominal model with a polynomial number of additional variables and constraints, so as to obtain its robust counterpart. Robustness can also be enforced by adding a possibly exponential family of cutting planes, which typically leads to an exponential formulation where cuts have to be generated at run time. Both approaches have pros and cons, and it is not clear which is the best one when approaching a specific problem. In this paper we computationally compare the two options on some prototype problems with different characteristics. We first address robust optimization a la Bertsimas and Sim for linear programs, and show through computational experiments that a considerable speedup (up to 2 orders of magnitude) can be achieved by exploiting a dynamic cut generation scheme. For integer linear problems, instead, the compact formulation exhibits a typically better performance. We then move to a probabilistic setting and introduce the uncertain set covering problem where each column has a certain probability of disappearing, and each row has to be covered with high probability. A related uncertain graph connectivity problem is also investigated, where edges have a certain probability of failure. For both problems, compact ILP models and cutting plane solution schemes are presented and compared through extensive computational tests. The outcome is that a compact ILP formulation (if available) can be preferable because it allows for a better use of the rich arsenal of preprocessing/cut generation tools available in modern ILP solvers. For the cases where such a compact ILP formulation is not available, as in the uncertain connectivity problem, we propose a restart solution strategy and computationally show its practical effectiveness.

Comparability Graph Coloring for Optimizing Utilization of Software-Managed Stream Register Files for Stream Processors

The stream processors represent a promising alternative to traditional cache-based general-purpose processors in achieving high performance in stream applications (media and some scientific applications). In a stream programming model for stream processors, an application is decomposed into a sequence of kernels operating on streams of data. During the execution of a kernel on a stream processor, all streams accessed must be communicated through a nonbypassing software-managed on-chip memory, the SRF (Stream Register File). Optimizing utilization of the scarce on-chip memory is crucial for good performance. The key insight is that the interference graphs (IGs) formed by the streams in stream applications tend to be comparability graphs or decomposable into a set of comparability graphs. We present a compiler algorithm for finding optimal or near-optimal colorings, that is, SRF allocations in stream IGs, by computing a maximum spanning forest of the sub-IG formed by long live ranges, if necessary. Our experimental results validate the optimality and near-optimality of our algorithm by comparing it with an ILP solver, and show that our algorithm yields improved SRF utilization over the First-Fit bin-packing algorithm, the best in the literature.

Diagnosing Multiple Byzantine Open-Segment Defects Using Integer Linear Programming

Faulty behaviors of open-segment defects are non-deterministic due to the Byzantine effect induced by the physical circuit layout. It is the test pattern and difficult for traditional ATPGs to manifest the corresponding faulty effect. Therefore, we propose a three-stage diagnosis approach for finding multiple open-segment defects. Stage one applies path tracing to help extract candidate fault sites from error outputs of failing patterns. An ILP solver in stage two effectively enumerates all fault combinations when considering fault candidates and simulation responses simultaneously. During stage three, fault simulation with support of physical information is responsible for identifying true open-segment defects by pruning false cases. Experimental results show good resolutions (only 1.7X and 1.5X total numbers of segments on average under 1,000 random and 5-detect patterns, respectively) for all ISCAS’85 circuits with 2–5 randomly-injected open-segment defects.

A set partitioning reformulation of a school bus scheduling problem

We present an integer programming model for the integrated optimization of bus schedules and school starting times, which is a single-depot vehicle scheduling problem with additional coupling constraints among the time windows. For instances with wide time windows the linear relaxation is weak and feasible solutions found by an ILP solver are of poor quality. We apply a set partitioning relaxation to compute better lower bounds and, in combination with a primal construction heuristic, also better primal feasible solutions. Integer programs with at most two non-zero coefficient per constraint play a prominent role in our approach. Computational results for several random and a real-world instance are given and compared with results from a standard branch-and-cut approach.

Lower bounds for the ITC-2007 curriculum-based course timetabling problem

This paper describes an approach for generating lower bounds for the curriculum-based course timetabling problem, which was presented at the International Timetabling Competition (ITC-2007, Track 3). So far, several methods based on integer linear programming have been proposed for computing lower bounds of this minimization problem. We present a new partition-based approach that is based on the “divide and conquer” principle. The proposed approach uses iterative tabu search to partition the initial problem into sub-problems which are solved with an ILP solver. Computational outcomes show that this approach is able to improve on the current best lower bounds for 12 out of the 21 benchmark instances, and to prove optimality for 6 of them. These new lower bounds are useful to estimate the quality of the upper bounds obtained with various heuristic approaches.

Computational experience with a core-based reduction procedure for the 2-knapsack problem

In this note we consider the 2-knapsack problem where each item has both a weight and a size. We show that a simple yet powerful reduction procedure can significantly enhance the capabilities of a commercial ILP solver, leading to solve very large instances.

Incremental demand rerouting: optimization models and algorithms

This investigation addresses an important and difficult combinatorial optimization problem in the design and management of telecommunication systems known as the path assignment problem. The path assignment problem is specified by a set of demands on a network with given link capacities. Each demand must be assigned to exactly one routing path in such a way that the total amount of traffic routed over any link does not exceed that link's capacity. Given a network and a path assignment, the problem of interest is to maximize the available bandwidth (throughput) for a new demand. Network managers are concerned with the quality of service provided by their networks, and so they are reluctant to make major changes to an operating network that may inadvertently result in service disruptions for numerous clients simultaneously. Therefore, the objective of this investigation is to develop and test optimization models and algorithms that produce a series of throughput-improving modifications to the original path assignment. This allows network managers to implement the improved routing in stages with minimal changes to the overall routing plan between any two consecutive stages. Although the procedure may only reroute a few demands at each stage, the routing after the last stage should be as close as possible to an optimal routing. That is, it must maximize the throughput for the new demand. We present integer programming models and associated solution algorithms to maximize the throughput with a sequence of incremental path modifications. The algorithms were implemented with the AMPL modeling language and the CPLEX ILP solver, and tested on a family of five different data sets based on a European network widely studied in the literature. Empirical results show that the incremental approach finds near-optimal results within reasonable limits on CPU time.

A computational evaluation of a general branch-and-price framework for capacitated network location problems

The purpose of this paper is to illustrate a general framework for network location problems, based on column generation and branch-and-price. In particular we consider capacitated network location problems with single-source constraints. We consider several different network location models, by combining cardinality constraints, fixed costs, concentrator restrictions and regional constraints. Our general branch-and-price-based approach can be seen as a natural counterpart of the branch-and-cut-based commercial ILP solvers, with the advantage of exploiting the tightness of the lower bound provided by the set partitioning reformulation of network location problems. Branch-and-price and branch-and-cut are compared through an extensive set of experimental tests.

A Local-Search-Based Heuristic for the Demand-Constrained Multidimensional Knapsack Problem

We consider an extension of the 0–1 multidimensional knapsack problem in which there are greater-than-or-equal-to inequalities, called demand constraints, in addition to the standard less-than-or-equal-to constraints. Moreover, the objective function coefficients are not constrained in sign. This problem is worth considering because it is embedded in models of practical application, it has an intriguing combinatorial structure, and it appears to be a challenging problem for commercial ILP solvers. Our approach is based on a nested tabu-search algorithm in which neighborhoods with different structures are exploited. First, a tabu-search procedure is carried out in which mainly the infeasible region is explored. Once feasibility has been established, a second tabu-search procedure, which analyzes only feasible solutions, is applied. The algorithm has been tested on a wide set of instances. Computational results are discussed.