Adjacency Variables Formulation for the Minimum Linear Arrangement Problem

The connection between linear and 0–1 integer linear formulations has attracted the attention of many researchers. The main reason triggering this interest has been an availability of efficient computer programs for solving pure linear problems including the transportation problem. Also the optimality of linear problems is easily verifiable through existing algorithms. However, there is no efficient general technique available to solve 0–1 integer linear problems or to verify their optimality. This paper shows that in the case of one of the easier 0–1 integer linear problems, namely a single assignment problem, such a relation between linear and 0–1 integer linear formulation can be built. The theory behind the proposed ‘bridge’ is based on the combination of the absolute point principle and shadow price theory. The main practical benefit of this work is in providing an algorithm to find a MFL (more-for-less) solution for the assignment problem. To the best of our knowledge, this is one of the first efforts to provide a ‘more-for-less’ result for a 0–1 integer linear problem.

Strategic bidding problems in energy markets have been vastly studied in the literature due to their economic importance and mathematical difficulty. In this paper, we consider this problem under the natural uncertainty associated to the competitors' bids and the system demand, and formulate it as a nonconvex quadratically constrained quadratic program. In order to solve this nonconvex maximization problem or to evaluate the quality of heuristic solutions, it is important to know good upper bounds to its optimal value. With this goal, we apply a cutting plane algorithm (CPA) to solve an extended linear relaxation of the problem, which is an outer approximation of a semidefinite relaxation. Valid inequalities are added to an initial linear relaxation at each iteration of the CPA improving the bound computed, including inequalities derived from reformulation linearization techniques (RLTs) and semidefinite programing (SDP) cuts that enforce the positive semidefiniteness of the matrix variable. SDP cuts are well known to considerably delay the solution of the relaxation because of their density, and therefore, we apply a sparsification procedure for the cuts, introduced by Margot et al. in . We propose a modification to the algorithm in Margot et al. to better control the density of the SDP cuts; and after tuning the parameters of our algorithm, we show the trade‐off between the bounds and the density of the cuts for test instances derived from the Brazilian electric system.

In this paper we study a version of the Multidimensional Arrangement Problem (MAP) that embeds a graph into a multidimensional array minimizing the aggregated (Manhattan) distance of the embedded edges. This problem includes the minimum Linear Arrangement Problem (minLA) as a special case, among others. We propose JAM, a tabu-based two-stage simulated annealing heuristic for this problem. Our algorithm relies on existing techniques for the minimum linear arrangement (minLA) problem, which are non-trivially adapted to work in multiple dimensions. Due to the scarcity of specific benchmarks for MAP, we have tested the performance of our algorithm with benchmarks for the minLA and Quadratic Assignment Problems (with more than 80 graphs). For each graph in these benchmarks, we provide results for 1, 2 and 3-dimensional instances of MAP, enlarging, hence, the benchmarking resources for the research community. The results obtained show the practicality of JAM, often matching the best known result and even improving some of them.

Parallel algorithms for the minimum cut and the minimum length tree layout problems

The node-edge weighted 2-edge connected subgraph problem: linear relaxation, facets and separation

Valid inequalities for the k-Color Shortest Path Problem

A deterministic global optimization algorithm for generalized geometric programming

This paper proposed an algorithm based on the branch-and-bound framework for globally solving the sum of linear ratios problem (SLRP) with a large number of ratios and a small number of variables. First, we introduced new variables to construct an equivalent problem of the problem (SLRP). Then, using a new linear relaxation technique, we obtained the linear relaxation problem for the equivalent problem. By utilizing the separable nature of the linear relaxation problem, we computed the linear relaxation problem by solving its $ p $ linear programming subproblems, thereby the lower bound for the problem (SLRP) could be obtained. Additionally, we conducted a theoretical analysis of the proposed algorithm and validated its feasibility and effectiveness through numerical experiments.

Partitioning a graph into balanced connected classes: Formulations, separation and experiments

This paper compares different exact approaches to solve the Discrete Ordered Median Problem (DOMP). In recent years, DOMP has been formulated using set packing constraints giving rise to one of its most promising formulations. The use of this family of constraints, known as strong order constraints (SOC), has been validated in the literature by its theoretical properties and because their linear relaxation provides very good lower bounds. Furthermore, embedded in branch-and-cut or branch-price-and-cut procedures as valid inequalities, they allow one to improve computational aspects of solution methods such as CPU time and use of memory. In spite of that, the above mentioned formulations require to include another family of order constraints, e.g., the weak order constraints (WOC), which leads to coefficient matrices with elements other than {0,1}. In this work, we develop a new approach that does not consider extra families of order constraints and furthermore relaxes SOC -in a branch-and-cut procedure that does not start with a complete formulation- to add them iteratively using row generation techniques to certify feasibility and optimality. Exhaustive computational experiments show that it is advisable to use row generation techniques in order to only consider {0,1}-coefficient matrices modeling the DOMP. Moreover, we test how to exploit the problem structure. Implementing an efficient separation of SOC using callbacks improves the solution performance. This allows us to deal with bigger instances than using fixed cuts/constraints pools automatically added by the solver in the branch-and-cut for SOC, concerning both the formulation based on WOC and the row generation procedure.

Optimization of the interaction distance between qubits to map a quantum circuit into one-dimensional quantum architectures is addressed. The problem is formulated as the Minimum Linear Arrangement (MinLA) problem. To achieve this, an interaction graph is constructed for a given circuit, and multiple instances of the MinLA problem for selected subcircuits of the initial circuit are formulated and solved. In addition, a lookahead technique is applied to improve the cost of the proposed solution which examines different subcircuit candidates. Experiments on quantum circuits for quantum Fourier transform and reversible benchmarks show the effectiveness of the approach.

The Group Problems and Subadditive Functions

In this paper, we study the efficiency (both theoretically and computationally) of a class of valid inequalities for the minimum weighted elementary directed cycle problem (MWEDCP) in planar digraphs with negative weight elementary directed cycles. These valid inequalities are called cycle valid inequalities and are parametrized by an integer called inequality’s order. From a theoretical point of view, we prove that separating cycle valid inequalities of order 1 in planar digraph can be done in polynomial time. From a computational point of view, we present a cutting plane algorithm featuring the efficiency of a lifted form of the cycle valid inequalities of order 1. In addition to these lifted valid inequalities, our algorithm is also based on a mixed integer linear formulation of the MWEDCP. The computational results are carried out on randomly generated planar digraph instances of the MWEDCP. For all 29 instances considered, we obtain in average 26.47% gap improvement. Moreover, for some of our instances the strengthening process directly displays the optimal integer elementary directed cycle.

Many safety-critical real-time embedded systems need to meet stringent timing constraints such as preserving delay bounds between input and output events. In model-based development, a system is often implemented by using a code generator to automatically generate source code from system models, and integrating the generated source code with a platform. It is challenging to guarantee that the implemented systems preserve required timing constraints, because the timed behavior of the source code and the platform is closely intertwined. In this paper, we address this challenge by proposing a model transformation approach for the code generation. Our approach compensates the platform-processing delays by adjusting the timing parameters in system models, based on an Integer Linear Programming problem formulation. We demonstrate the usefulness of our approach via a case study of infusion pump systems. Experimental results show that the code generated using our approach can better preserve the timing constraints.

In this paper, a branch and bound approach is proposed for global optimization problem (P) of the sum of generalized polynomial fractional functions under generalized polynomial constraints, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. By utilizing an equivalent problem and some linear underestimating approximations, a linear relaxation programming problem of the equivalent form is obtained. Consequently, the initial non-convex nonlinear problem (P) is reduced to a sequence of linear programming problems through successively refining the feasible region of linear relaxation problem. The proposed algorithm is convergent to the global minimum of the primal problem by means of the solutions to a series of linear programming problems. Numerical results show that the proposed algorithm is feasible and can successfully be used to solve the present problem (P).