A polynomial algorithm for integer programming covering problems satisfying the integer round-up property

LetA be a non-negative matrix with integer entries and no zero column. The integer round-up property holds forA if for every integral vectorw the optimum objective value of the generalized covering problem min{1y: yA w, y 0 integer} is obtained by rounding up to the nearest integer the optimum objective value of the corresponding linear program. A polynomial time algorithm is presented that does the following: given any generalized covering problem with constraint matrixA and right hand side vectorw, the algorithm either finds an optimum solution vector for the covering problem or else it reveals that matrixA does not have the integer round-up property.     

The C 3 Theorem and a D 2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification

This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C³ (Common Cut Coefficients) Theorem. Based on the C³ Theorem, we develop a decomposition algorithm which we refer to as Disjunctive Decomposition (D²). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D² algorithm.This research was funded by NSF grants DMII 9978780 and CISE 9975050.     

Global minimization of large-scale constrained concave quadratic problems by separable programming

The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variablesx R ⁿ , while the linear part is in terms ofy R ^k. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. Ana priori bound on this relative error is obtained, which is shown to be 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteed-approximate solution is obtained by solving a single liner zero–one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem.This research was supported by the Division of Computer Research, National Science Foundation under Research Grant DCR8405489.Dedicated to Professor George Dantzig in honor of his 70th Birthday.     

A o(n logn) algorithm for LP knapsacks with GUB constraints

A specialization of the dual simplex method is developed for solving the linear programming (LP) knapsack problem subject to generalized upper bound (GUB) constraints. The LP/GUB knapsack problem is of interest both for solving more general LP problems by the dual simplex method, and for applying surrogate constraint strategies to the solution of 0–1 Multiple Choice integer programming problems. We provide computational bounds for our method of o(n logn), wheren is the total number of problem variables. These bounds reduce the previous best estimate of the order of complexity of the LP/GUB knapsack problem (due to Witzgall) and provide connections to computational bounds for the ordinary knapsack problem.We further provide a variant of our method which has only slightly inferior worst case bounds, yet which is ideally suited to solving integer multiple choice problems due to its ability to post-optimize while retaining variables otherwise weeded out by convex dominance criteria.     

Sensitivity theorems in integer linear programming

We consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right-hand-side vector. Among our results, we show that, for any optimal solution to a linear program max{wx: Axb}, the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixA. Using this, we strengthen several integer programming proximity results of Blair and Jeroslow; Graver; and Wolsey. We also show that the Chvátal rank of a polyhedron {x: Axb} can be bounded above by a function of the matrixA, independent of the vectorb, a result which, as Blair observed, is equivalent to Blair and Jeroslow's theorem that each integer programming value function is a Gomory function.Supported by a grant from the Alexander von Humboldt Stiftung.Since September 1985: Department of Operations Research, Upson Hall, Cornell University, Ithaca, NY 14853, USA.Partially supported by the Sonderforschungbereich 21 (DFG), Institut für Ökonometrie und Operations Research of the University of Bonn, FR Germany.     

Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming

Decomposition has proved to be one of the more effective tools for the solution of large-scale problems, especially those arising in stochastic programming. A decomposition method with wide applicability is Benders' decomposition, which has been applied to both stochastic programming as well as integer programming problems. However, this method of decomposition relies on convexity of the value function of linear programming subproblems. This paper is devoted to a class of problems in which the second-stage subproblem(s) may impose integer restrictions on some variables. The value function of such integer subproblem(s) is not convex, and new approaches must be designed. In this paper, we discuss alternative decomposition methods in which the second-stage integer subproblems are solved using branch-and-cut methods. One of the main advantages of our decomposition scheme is that Stochastic Mixed-Integer Programming (SMIP) problems can be solved by dividing a large problem into smaller MIP subproblems that can be solved in parallel. This paper lays the foundation for such decomposition methods for two-stage stochastic mixed-integer programs.     

The polynomial hierarchy and a simple model for competitive analysis

The multi-level linear programs of Candler, Norton and Townsley are a simple class of sequenced-move games, in which players are restricted in their moves only by common linear constraints, and each seeks to optimize a fixed linear criterion function in his/her own continuous variables and those of other players. All data of the game and earlier moves are known to a player when he/she is to move. The one-player case is just linear programming.We show that questions concerning only the value of these games exhibit complexity which goes up all levels of the polynomial hierarchy and appears to increase with the number of players.For three players, the games allow reduction of the ₂ and ₂ levels of the hierarchy. These levels essentially include computations done with branch-and-bound, in which one is given an oracle which can instantaneously solve NP-complete problems (e.g., integer linear programs). More generally, games with (p + 1) players allow reductions of _p and _p in the hierarchy.An easy corollary of these results is that value questions for two-player (bi-level) games of this type is NP-hard.The author's work has been supported by the Alexander von Humboldt Foundation and the Institut fur Okonometrie und Operations Research of the University of Bonn, Federal Republic of Germany; grant ECS8001763 of the National Science Foundation, USA; and a grant from the Georgia Tech Foundation.     

Penalty for zero–one integer equivalent problem

The equivalence of zero–one integer programming and a concave quadratic penalty function problem has been shown by Raghavachari, for a sufficiently large value of the penalty. A lower bound for this penalty is obtained here, which in specific cases cannot be reduced.This research was supported in part by the Computer Science Section of the National Science Foundation under Research Grant MCS 8101214.     

Connections between the Support and Linear Independence of Refinable Distributions

The purpose of this paper is to study the relationships between the support of a refinable distributionφand the global and local linear independence of the integer translates ofφ. It has been shown elsewhere that a compactly supported distributionφhas globally independent integer translates if and only ifφhas minimal convex support. However, such a distribution may have “holes” in its support. By insisting thatφ∈L²() and generates a multiresolution analysis, Lemarié and Malgouyres have ensured that no such holes can occur. In this article we generalize this result to refinable distributions. We also give a result on the local linear independence of the integer translates ofφ. We work with integer dilation factorN?2 throughout this paper.     

Implicit representation of generalized variable upper bounds in linear programming

In certain linear programs, especially those derived from integer programs, large numbers of constraints may have very simple form. Examples are:x _ij 1 (simple upper bounds [SUB]), _i x _ij = 1 (generalized upper bounds [GUB]) andx _ij y _i (variable upper bounds [VUB]). A class of constraints called generalized VUB [GVUB] is introduced which includes GUB and VUB as special cases. Also introduced is a method for representing GVUB constraints implicitly within the mechanics of the simplex method.Research supported in part by the Mobil Oil Corporation.     

Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes

The existence problem for a semicyclic group divisible design (SCGDD) of type m ⁿ with block size 4 and index unity, denoted by 4-SCGDD, has been studied for any odd integer m to construct a kind of two-dimensional optical orthogonal codes (2-D OOCs) with the AM-OPPW (at most one-pulse per wavelength) restriction. In this paper, the existence of a 4-SCGDD of type m ⁿ is determined for any even integer m, with some possible exceptions. A complete asymptotic existence result for k-SCGDDs of type m ⁿ is also obtained for all larger k and odd integer m. All these SCGDDs are used to derive new 2-D OOCs with the AM-OPPW restriction, which are optimal in the sense of their sizes.     

A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (ℓ,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (ℓ,S) inequalities to a general class of valid inequalities, called the inequalities, and we establish necessary and sufficient conditions which guarantee that the inequalities are facet-defining. A separation heuristic for inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the inequalities as cuts. This research has been supported in part by the National Science Foundation under Award number DMII-0121495.     

On binary quadratic forms with the semigroup property

A quadratic form f is said to have the semigroup property if its values at the points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with the semigroup property. If there is an integer bilinear map s such that f(s(x,y)) = f(x)f(y) for all vectors x and y from the integer two-dimensional lattice, then the form f has the semigroup property. We give an explicit integer parameterization of all pairs (f,s) with the property stated above. We do not know any other examples of forms with the semigroup property.     

Simplex pivots on the set packing polytope

This paper considers the set packing problem max{wx: Ax b, x 0 and integral}, whereA is anm × n 0–1 matrix,w is a 1 ×n weight vector of real numbers andb is anm × 1 vector of ones. In equality form, its linear programming relaxation is max{wx: (x, y) P(A)} whereP(A) = {(x, y):Ax +I _m y =b, x0,y0}. Letx ¹ be any feasible solution to the set packing problem that is not optimal and lety ¹ =b – Ax ¹; then (x ¹,y ¹) is an integral extreme point ofP(A). We show that there exists a sequence of simplex pivots from (x ¹,y ¹) to (x*,y*), wherex* is an optimal solution to the set packing problem andy* =b – Ax*, that satisfies the following properties. Each pivot column has positive reduced weight and each pivot element equals plus one. The number of pivots equals the number of components ofx* that are nonbasic in (x ¹,y ¹).This research was supported by NSF Grants ECS-8005360 and ECS-8307473 to Cornell University.     

A Theorem Concerning the Integer Lattice

This note shows that if an integer linear program in n variables has more than 2ⁿ linear inequality constraints, then either some of the constraints are unnecessary or there is at least one feasible integer point.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.     

Extended integer rank reduction formulas and Smith normal form

We present an integer rank reduction formula for transforming the rows and columns of an integer matrix A. By repeatedly applying the formula to reduce rank, an extended integer rank reducing process is derived. The process provides a general finite iterative approach for constructing factorizations of A and A ^T under a common framework of a general decomposition V ^T AP?=?Ω. Then, we develop the integer Wedderburn rank reduction formula and its integer biconjugation process. Both the integer biconjugation process associated with the Wedderburn rank reduction process and the scaled extended integer Abaffy–Broyden–Spedicato (ABS) class of algorithms are shown to be in the integer rank reducing process. We also show that the integer biconjugation process can be derived from the scaled integer ABS class of algorithms applied to A or A ^T . Finally, we show that the integer biconjuagation process is a special case of our proposed ABS class of algorithms for computing the Smith normal form.     

Equiangular lines in C (part II)

A subset S of a complex projective space is F-regular provided each two points of S have the same non-zero distance and each subset of three points of S has the same shape invariant. The aim of this paper is the determination for any odd integer r, of the largest integer n(r) such tht CP^r−1 contains an F-regular subset of n(r) points.It is established that n(r) ≤ 2r − 2 for any odd integer r and n(1 + 2^s) = 2^s+1 for any integer s.     

Integer Optimization on Convex Semialgebraic Sets

Let Y be a convex set in \Re ^k defined by polynomial inequalities and equations of degree at most d ≥ 2 with integer coefficients of binary length at most l . We show that if the set of optimal solutions of the integer programming problem min is not empty, then the problem has an optimal solution of binary length ld ^O(k4) . For fixed k , our bound implies a polynomial-time algorithm for computing an optimal integral solution y ^* . In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming. Received August 3, 1998, and in revised form March 22, 1999.     

Typical primitive polynomials over integer residue rings

Let N be a product of distinct prime numbers and Z/(N) be the integer residue ring modulo N. In this paper, a primitive polynomial f(x) over Z/(N) such that f(x) divides x^s−c for some positive integer s and some primitive element c in Z/(N) is called a typical primitive polynomial. Recently typical primitive polynomials over Z/(N) were shown to be very useful, but the existence of typical primitive polynomials has not been fully studied. In this paper, for any integer m1, a necessary and sufficient condition for the existence of typical primitive polynomials of degree m over Z/(N) is proved.