A pegging approach to the precedence-constrained knapsack problem

The knapsack problem (KP) is generalized to the case where items are partially ordered through a set of precedence relations. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be packed in the knapsack. However, each item can be accepted only when all the preceding items have been included in the knapsack. The knapsack problem with these additional constraints is referred to as the precedence-constrained knapsack problem (PCKP). To solve PCKP exactly, we present a pegging approach, where the size of the original problem is reduced by applying the Lagrangian relaxation followed by a pegging test. Through this approach, we are able to solve PCKPs with thousands of items within a few minutes on an ordinary workstation.     

Lifting valid inequalities for the precedence constrained knapsack problem

This paper considers the precedence constrained knapsack problem. More specifically, we are interested in classes of valid inequalities which are facet-defining for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities arising from minimal induced covers, we identify a class of lifting coefficients for which this problem can be solved in polynomial time, by using a supermodular function, and for which the values of the lifting coefficients have a combinatorial interpretation. For the remaining lifting coefficients it is shown that this optimization problem is strongly NP-hard. The same lifting procedure can be applied to (1,k)-configurations, although in this case, the same combinatorial interpretation no longer applies. We also consider K-covers, to which the same procedure need not apply in general. We show that facets of the polytope can still be generated using a similar lifting technique. For tree knapsack problems, we observe that all lifting coefficients can be obtained in polynomial time. Computational experiments indicate that these facets significantly strengthen the LP-relaxation. Received July 10, 1997 / Revised version received January 9, 1999? Published online May 12, 1999     

Heuristic and Exact Algorithms for the Precedence-Constrained Knapsack Problem

The knapsack problem (KP) is generalized taking into account a precedence relation between items. Such a relation can be represented by means of a directed acyclic graph, where nodes correspond to items in a one-to-one way. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be included in the knapsack. However, each item can be adopted only when all of its predecessors have been included in the knapsack. The knapsack problem with such an additional set of constraints is referred to as the precedence-constrained knapsack problem (PCKP). We present some dynamic programming algorithms that can solve small PCKPs to optimality, as well as a preprocessing method to reduce the size of the problem. Combining these, we are able to solve PCKPs with up to 2000 items in less than a few minutes of CPU time.     

A cutting plane algorithm for the capacitated facility location problem

The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of clients. A multi-source version of the problem is considered in which each client can be served by more than one facility. In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable. We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope. We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable. To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach.     

n-step mingling inequalities: new facets for the mixed-integer knapsack set

The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313–346, 2009) are valid inequalities for the mixed-integer knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315–338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into n-step MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n?=?2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that n-step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets.     

Mingling: mixed-integer rounding with bounds

Mixed-integer rounding (MIR) is a simple, yet powerful procedure for generating valid inequalities for mixed-integer programs. When used as cutting planes, MIR inequalities are very effective for mixed-integer programming problems with unbounded integer variables. For problems with bounded integer variables, however, cutting planes based on lifting techniques appear to be more effective. This is not surprising as lifting techniques make explicit use of the bounds on variables, whereas the MIR procedure does not. In this paper we describe a simple procedure, which we call mingling, for incorporating variable bound information into MIR. By explicitly using the variable bounds, the mingling procedure leads to strong inequalities for mixed-integer sets with bounded variables. We show that facets of mixed-integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure. In particular, the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of Marchand and Wolsey (Math Program 85:15–33, 1999) as well as the continuous integer knapsack cover and pack inequalities of Atamtürk (Math Program 98:145–175, 2003; Ann Oper Res 139:21–38, 2005). In addition, mingling inequalities give a generalization of the two-step MIR inequalities of Dash and Günlük (Math Program 105:29–53, 2006) under some conditions.     

The piecewise linear optimization polytope: new inequalities and intersection with semi-continuous constraints

We give new facets and valid inequalities for the separable piecewise linear optimization (SPLO) knapsack polytope. We also extend the inequalities to the case in which some of the variables are semi-continuous. Finally, we give computational results that demonstrate their efficiency in solving difficult instances of SPLO and SPLO with semi-continuous constraints.     

A polyhedral study of the cardinality constrained knapsack problem

 A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas such as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems. Received: March 13, 2003 Published online: April 10, 2003 Key Words. mixed-integer programming – knapsack problem – cardinality constrained programming – branch-and-cut     

Second-order cover inequalities

We introduce a new class of second-order cover inequalities whose members are generally stronger than the classical knapsack cover inequalities that are commonly used to enhance the performance of branch-and-cut methods for 0–1 integer programming problems. These inequalities result by focusing attention on a single knapsack constraint in addition to an inequality that bounds the sum of all variables, or in general, that bounds a linear form containing only the coefficients 0, 1, and –1. We provide an algorithm that generates all non-dominated second-order cover inequalities, making use of theorems on dominance relationships to bypass the examination of many dominated alternatives. Furthermore, we derive conditions under which these non-dominated second-order cover inequalities would be facets of the convex hull of feasible solutions to the parent constraints, and demonstrate how they can be lifted otherwise. Numerical examples of applying the algorithm disclose its ability to generate valid inequalities that are sometimes significantly stronger than those derived from traditional knapsack covers. Our results can also be extended to incorporate multiple choice inequalities that limit sums over disjoint subsets of variables to be at most one.      

Valid inequalities and facets of the capacitated plant location problem

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure.The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with a constant capacity for all plants. These facet inequalities depend on the capacity and thus differ fundamentally from the valid inequalities for the uncapacited version of the problem.We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.This research was partially supported by Grant # ECS-8316224 from the National Science Foundation's Program in Systems Theory and Operations Research.     

Solving a gas-lift optimization problem by dynamic programming

Gas lift is a costly, however indispensable means to recover oil from high-depth reservoirs that entails solving the gas-lift optimization problem, GOP, often in response to variations in the dynamics of the reservoir and economic oscillations. GOP can be cast as a mixed integer nonlinear programming problem whose integer variables decide which oil wells should produce, while the continuous variables allocate the gas-compressing capacity to the active ones. This paper extends the GOP formulation to encompass uncertainties in the oil outflow and precedence constraints imposed on the activation of wells. Recursive solutions are suggested for instances devoid of precedence constraints, as well as instances arising from precedence constraints induced by forests and general acyclic graphs. For the first two classes, pseudo-polynomial algorithms are developed to solve a discretized version of GOP, while the most general version is shown to be NP-Hard in the strong sense. Numerical experiments provide evidence that the approximate algorithms obtained by solving the recurrences produce near-optimal solutions. Further, the family of cover inequalities of the knapsack problem is extended to the gas-lift optimization problem.     

Cyclic group and knapsack facets

 Any integer program may be relaxed to a group problem. We define the master cyclic group problem and several master knapsack problems, show the relationship between the problems, and give several classes of facet-defining inequalities for each problem, as well as a set of mappings that take facets from one type of master polyhedra to another. Received: May 24, 2001 / Accepted: August 2002 Published online: March 21, 2003 Mathematics Subject Classification (1991): 20E28, 20G40, 20C20     

On a generalization of the master cyclic group polyhedron

We study the master equality polyhedron (MEP) which generalizes the master cyclic group polyhedron (MCGP) and the master knapsack polyhedron (MKP). We present an explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP. This result generalizes similar results for the MCGP by Gomory (1969) and for the MKP by Araóz (1974). Furthermore, this characterization gives a polynomial time algorithm for separating an arbitrary point from MEP. We describe how facet-defining inequalities for the MCGP can be lifted to obtain facet-defining inequalities for MEP, and also present facet-defining inequalities for MEP that cannot be obtained in such a way. Finally, we study the mixed-integer extension of MEP and present an interpolation theorem that produces valid inequalities for general mixed integer programming problems using facets of MEP.     

Enhancing discretized formulations: the knapsack reformulation and the star reformulation

Discretized formulations have proved to be useful for modeling combinatorial optimizations. The main focus of this work is on how to strengthen the linear programming relaxation of a given discretized formulation. More precisely, we will study and strengthen subproblems that arise in these formulations. In one case we will focus on the so-called knapsack reformulation which is based on viewing these models as the intersection of two polyhedra, one of them being a specialized knapsack problem. We will show that strong inequalities used in previous works are a special case of inequalities implied by the knapsack formulation. In the second case we will analyze a star-like subproblem and will provide an extended formulation for this problem as well as a set of inequalities on the original space, implied by the inequalities of the extended formulation. We will use a generalization of the Degree Constrained Spanning Tree problem as a setting for this study. In the present work, besides contextualizing these enhancements in terms of discretized models presented in previous works, we also compare and combine together them, for the first time.     

Lifted inequalities for 0-1 mixed integer programming: Superlinear lifting

We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables, through the lifting of continuous variables fixed at their upper bounds. We introduce the concept of a superlinear inequality and show that, in this case, lifting is significantly simpler than for general inequalities. We use the superlinearity theory, together with the traditional lifting of 0-1 variables, to describe families of facets of the mixed 0-1 knapsack polytope. Finally, we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large.This research was supported by NSF grants DMI-0100020 and DMI-0121495Mathematics Subject Classification (1991): 90C11, 90C27     

Dynamic knapsack sets and capacitated lot-sizing

 A dynamic knapsack set is a natural generalization of the 0-1 knapsack set with a continuous variable studied recently. For dynamic knapsack sets a large family of facet-defining inequalities, called dynamic knapsack inequalities, are derived by fixing variables to one and then lifting. Surprisingly such inequalities have the simultaneous lifting property, and for small instances provide a significant proportion of all the facet-defining inequalities. We then consider single-item capacitated lot-sizing problems, and propose the joint study of three related sets. The first models the discrete lot-sizing problem, the second the continuous lot-sizing problem with Wagner-Whitin costs, and the third the continuous lot-sizing problem with arbitrary costs. The first set that arises is precisely a dynamic knapsack set, the second an intersection of dynamic knapsack sets, and the unrestricted problem can be viewed as both a relaxation and a restriction of the second. It follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets. Some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot-sizing problems. Received: March 28, 2001 / Accepted: November 9, 2001 Published online: September 27, 2002 RID="★" ID="★" Research carried out with financial support of the project TMR-DONET nr. ERB FMRX–CT98–0202 of the European Union. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Key words. knapsack sets – valid inequalities – simultaneous lifting – lot-sizing – Wagner-Whitin costs     

Lifting the facets of zero–one polytopes

We discuss a procedure to obtain facets and valid inequalities for the convex hull of the set of solutions to a general zero–one programming problem. Basically, facets and valid inequalities defined on lower dimensional subpolytopes are lifted into the space of the original problem. The procedure generalizes the previously known techniques for lifting facets in two respects. First, the general zero–one programming problem is considered rather than various special cases. Second, the procedure is exhaustive in the sense that it accounts for all the facets (valid inequalities) which are liftings of a given lower dimensional facet (valid inequality).     

Capacitated facility location: Separation algorithms and computational experience

We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack cover, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NP-hard in general. For the well known subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set and general capacities, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining. The way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 medium size problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

The multi-item capacitated lot-sizing problem with setup times and shortage costs

We address a multi-item capacitated lot-sizing problem with setup times and shortage costs that arises in real-world production planning problems. Demand cannot be backlogged, but can be totally or partially lost. The problem is NP-hard. A mixed integer mathematical formulation is presented. Our approach in this paper is to propose some classes of valid inequalities based on a generalization of Miller et al. [A.J. Miller, G.L. Nemhauser, M.W.P. Savelsbergh, On the polyhedral structure of a multi-item production planning model with setup times, Mathematical Programming 94 (2003) 375–405] and Marchand and Wolsey [H. Marchand, L.A. Wolsey, The 0–1 knapsack problem with a single continuous variable, Mathematical Programming 85 (1999) 15–33] results. We also describe fast combinatorial separation algorithms for these new inequalities. We use them in a branch-and-cut framework to solve the problem. Some experimental results showing the effectiveness of the approach are reported.     

Cyclic group blocking polyhedra

Gomory (Linear Algebra Appl 2:451–558, 1969) gave a subadditive characterization of the facets of the group polyhedron. Although there are exponentially many facets (see Gomory and Johnson in Math Program 3:359–389, 1972, Example 4.6) and exponentially many vertices for the group polyhedron for the master cyclic group problem, Gomory’s characterization of the non-trivial facets has polynomially many subadditive inequalities, in fact of order |G|² for a finite Abelian group G. We reduce this subadditive inequality system to its minimal representation by a triple system of the same order and show the dimensionality of the polytope of non-trivial facets. The system of all triples corresponds to all solution vectors of length three into which every solution vector can be decomposed. Our minimal representation leads to a characterization of the vertices of length three. Gomory et al. (Math Program 96:321–339, 2003) introduced a shooting experiment involving solving the shooting linear program repeatedly to find important facets. We develop a topological network flow model of the dual problem of the shooting linear program in a reverse procedure from the decomposition of solution vectors into triples. Hunsaker (2003) gave a knapsack shooting experiment, which we use to find a simple pattern for the most hit knapsack facets.