Integer Programming Feasibility Research Articles

On the optimality of pseudo-polynomial algorithms for integer programming

In the classic Integer Programming Feasibility (IPF) problem, the objective is to decide whether, for a given \(m \times n\) matrix A and an m-vector \(b=(b_1,\dots , b_m)\), there is a non-negative integer n-vector x such that \(Ax=b\). Solving (IPF) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input. The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IPF) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ITCS 2019]. Jansen and Rohwedder designed an algorithm for (IPF) with running time \({\mathcal {O}}(m \varDelta )^m\) \(\log (\varDelta ) \log (\varDelta +\Vert b\Vert _{\infty })+{\mathcal {O}}(mn)\). Here, \(\varDelta \) is an upper bound on the absolute values of the entries of A. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal, by proving a lower bound of \(n^{o(\frac{m}{\log m})} \cdot \Vert b\Vert _{\infty }^{o(m)}\). We also prove that assuming ETH, (IPF) cannot be solved in time \(f(m)\cdot (n \cdot \Vert b\Vert _{\infty })^{o(\frac{m}{\log m})}\) for any computable function f. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IPF) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IPF) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IPF) when the path-width of the corresponding column-matroid is a constant .

Superadditive characterizations of pure integer programming feasibility

We derive a new family of integer programming (IP) Farkas’ lemmas based on superadditive duality. We construct a certificate of infeasibility for an IP as a function of its right-hand side. We develop a recursive approach to compute such a function.

Implementation of scheduling algorithm for optimisation by Lagrangian decomposition model

This paper mainly deals with the design and analysis of Lagrangian model using cyclic scheduling algorithm. An integer programming (IP) model is formulated and then decomposed by using Lagrangian relaxation technique. Two approaches were implemented, the first approach is based on the relaxation of the preference constraints and the second approach is based on the relaxation of the demand constraints. Since proper bounds were not obtained by relaxing the preference constraints, a solution methodology is developed based on the relaxation of demand constraints. Many researchers have discussed the different type of methods, such as bundle method, heuristic and variable fixing. A duality gap exists between the lower bound and the objective function value provided by an IP feasibility heuristic. A variable fixing scheme is introduced for the convergence criteria. Based on the numerical calculations and graphical representations, the minimum cost is obtained to cover the uncovered shifts.

Variable Neighbourhood Pump Heuristic for 0-1 Mixed Integer Programming Feasibility

Abstract In this paper we propose a new method for finding an initial feasible solution for Mixed integer programs. We call it “Variable neighborhood pump”, since it combines ideas of Variable neighborhood branching and Feasibility pump heuristics. The proposed heuristic was tested on an established set of 83 benchmark problems proven to be difficult to solve to feasibility. The results are compared with those of IBM ILOG CPLEX 11.1, which already includes standard feasibility pump as a primal heuristic. With our approach we managed to obtain better initial objective function values than CPLEX on 63 test instances, within similar average computational time.

Formal verification of sequence controllers

Sequence controllers are widely used in the chemical processing industries due to the inherently sequential nature of many process operations. In particular, start-up and shut-down operations in continuous processes and any batch operation require sequence controls to force the time-dependent progression of the operation. One incorrect sequence embedded in a sequence control system can potentially cause severe problems. Therefore, all sequences embedded in a sequence control system need to be checked for consistency with design specifications. A formal verification methodology is developed that can systematically verify the functionality of sequence control systems represented at the logic level. Our approach is based on extensions of the recently developed implicit model checking technology, which is particularly well suited to the verification of large and complex systems. The sequence control system is represented implicitly as a system of Boolean equations, and the sequences to be verified are specified formally with temporal logic. Formal verification then requires the solution of a series of Boolean satisfiability problems, which are solved efficiently as integer programming feasibility problems. The methodology is applied to a simplified sequence control system to illustrate its application during the design of sequence control systems. Finally, the methodology is applied to an existing industrial burner management system.

Implicit model checking of logic‐based control systems

AbstractIncreasing automation in the CPI has led to the growing use of logic‐based control systems (or safety interlock systems) in safety‐related and sequencing operations. The growing complexity and safety‐critical nature of typical applications make developing technologies for the formal verification of logic‐based control systems with respect to their functionality a crucial and urgent issue. It still remains elusive to analysis, primarily due to the exponential growth of the alternatives that must be examined with application size. Implicit model checking is a formal verification technology that can be applied to the verification of large‐scale logic‐based control system. Its primary advantage is to formally verify large‐scale coupled systems, where a novel and compact model formulation makes tractable previously inaccessible problems. Logic‐based control systems are represented compactly as an implicit Boolean state‐space model, and the properties to be verified are represented in the language of temporal logic. Verification is posed as a Boolean satisfiability problem and then transformed into its equivalent integer programming feasibility problem, which allows for efficient solution with standard branch‐and‐bound algorithms. Benefits of the methodology are demonstrated by applying to largescale industrial examples.

A continuous approach to inductive inference

In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean functionź:{0, 1}n ź {0, 1} using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input--output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used for defining the search region is dynamically modified. Computational results on 8-, 16- and 32-input, 1-output functions are presented. Our implementation successfully identified the majority of hidden functions in the experiment.

Decomposing finitely generated integral monoids by elimination

A finitely generated integral monoid can always be decomposed into a finite (although generally exponential) number of integral monoids having special structure, in that their related integer programming feasibility problems can be solved in polynomial time. The decomposition can be constructed using an elimination scheme due to Presburger and Williams that generalizes Fourier-Motzkin elimination.

An interior point algorithm to solve computationally difficult set covering problems

We present an interior point approach to the zero–one integer programming feasibility problem based on the minimization of a nonconvex potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such that each consecutive point reduces the value of the potential function. An integer solution (not necessarily feasible) is generated at each iteration by a rounding scheme. The direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid. We illustrate the approach by considering a class of difficult set covering problems that arise from computing the 1-width of the incidence matrix of Steiner triple systems.

The structure of an integral monoid and integer programming feasibility

Testing an integer programming problem for feasibility is equivalent to testing whether or not the right-hand side belongs to a certain integral monoid. Results on the structure of an integral monoid are discussed in relation to the integer programming feasibility problem.