Knapsack Constraints Research Articles

Solving a class of multiplicative programs with 01 knapsack constraints

Solving a class of multiplicative programs with 01 knapsack constraints

On surrogating 0–1 knapsack constraints

In this note, we present a scheme for tightening 0–1 knapsack constraints based on other knapsack constraints surrogating.

On using an automatic scheme for obtaining the convex hull defining inequalities of a Weismantel 0–1 knapsack constraint

In this short note we obtain the full set of inequalities that define the convex hull of a 0–1 knapsack constraint presented in Weismantel (1997). For that purpose we use our O(n) procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates, as well as our schemes for coefficient increase based tightening cover induced inequalities and coefficient reduction based tightening general 0–1 knapsack constraints.

Exact Solution of the Quadratic Knapsack Problem

The Quadratic Knapsack Problem (QKP) calls for maximizing a quadratic objective function subject to a knapsack constraint, where all coefficients are assumed to be nonnegative and all variables are binary. The problem has applications in location and hydrology, and generalizes the problem of checking whether a graph contains a clique of a given size. We propose an exact branch-and-bound algorithm for QKP, where upper bounds are computed by considering a Lagrangian relaxation that is solvable through a number of (continuous) knapsack problems. Suboptimal Lagrangian multipliers are derived by using subgradient optimization and provide a convenient reformulation of the problem. We also discuss the relationship between our relaxation and other relaxations presented in the literature. Heuristics, reductions, and branching schemes are finally described. In particular, the processing of each node of the branching tree is quite fast: We do not update the Lagrangian multipliers, and use suitable data structures to compute an upper bound in linear expected time in the number of variables. We report exact solution of instances with up to 400 binary variables, i.e., significantly larger than those solvable by the previous approaches. The key point of this improvement is that the upper bounds we obtain are typically within 1% of the optimum, but can still be derived effectively. We also show that our algorithm is capable of solving reasonable-size Max Clique instances from the literature.

O( n log n) procedures for tightening cover inequalities

We present two procedures for tightening cover induced inequalities in 0-1 programs by using knapsack constraints plus some other information from the program. The tightening is obtained by solving successive knapsack problems with all 0-1 objective function coefficients, and using dominance criteria to avoid the explicit solving of some knapsack problems. The new constraints are 0-1 equivalent to and LP tighter than the original ones. Both procedures have O (n log n) complexity, where n is the number of variables with nonzero coefficients in the knapsack constraint, however one of them can strongly reduce the computational effort. Some computational experience is reported.

Solving the generalised assignment problem using polyhedral results

The Generalised Assignment Problem (GAP) consists of finding a maximal profit assignment of n tasks over m capacity constrained agents, whereby each task has to be processed by only one agent. We develop an improved implementation of the standard procedure for generating lifted cover inequalities describing an approximation to the convex hull of the knapsack constraints in the GAP polytope. This improvement yields a good upper bound and closes the gap by an additional 15% on average. Based on this result, we propose two heuristics for finding close-to-optimal solutions, giving us a tight lower bound. Computational results on a set of 60 representative and highly capacitated problems indicate that these solutions lie within 0.06% of the optimum. After applying some pre-processing techniques and using the obtained bounds, we solve the generated instances to optimality by Branch-and-Bound (B&B) within reasonable computing time.

Solving a Truck Dispatching Scheduling Problem Using Branch-and-Cut

A branch-and-cut IP solver is developed for a class of structured 0/1 integer programs arising from a truck dispatching scheduling problem. This problem involves a special class of knapsack equality constraints. Families of facets for the polytopes associated with individual knapsack constraints are identified. In addition, a notion of “conflict graph” is utilized to obtain an approximating node-packing polytope for the convex hull of all 0/1 solutions. The branch-and-cut solver generates cuts based on both the knapsack equality constraints and the approximating node-packing polytope, and incorporates these cuts into a tree-search algorithm that uses problem reformulation and linear programming-based heuristics at each node in the search tree to assist in the solution process. Numerical experiments are performed on large-scale real instances supplied by Texaco Trading & Transportation, Inc. The optimal schedules correspond to cost savings for the company and greater job satisfaction for drivers due to more balanced work schedules and income distribution.

On characterizing tighter formulations for 0–1 programs

Abstract We present two approaches for 0–1 program model tightening. They are based on increasing and reducing knapsack constraint coefficients, respectively. Both approaches make use of information from cover structures implied by the original constraint system. The formulations that they provide are 0–1 equivalent and as tight as the original model. We also present some characterizations for the new formulations to be tighter than the original model.

On tightening 0-1 programs based on extensions ofpure 0-1 knapsack and subset-sum problems

We present a framework for automatic tightening of general 0-1 programs. A givenconstraint is tightened by using its own structure as well as information from the other constraints.Our approach exploits special structures that are frequently encountered in practicalproblems, namely knapsack constraints, cliques, covers, variable covers and variable upperbounds. We consider 0-1 knapsack and subset-sum problems with clique and cover inducedconstraints. The tightening (reduction and increasing) of constraint coefficients benefits fromimplication results due to probing analysis. Some computational experience is reported onwell-known real-life problems as well as on a new set of large-scale problems from theinvestment planning sector.

Assigning jobs to time frames on a single machine to minimize total tardiness

This paper deals with a sequencing problem arising in the management of paced-flowlines, that is production lines where jobs are released at constant time intervals. The problem is to sequence jobs to minimis total tardiness. The problem can be formulated as an assignment problem with a number of knapsack constraints. We prove the strong NP-hardness of the problem and give a number of lower bounds which are used in a branch-and-bound algorithm. Computational results in realistic settings confirm the effectiveness of the procedure developed. The results are particularly interesting with reference to mixed-model assembly lines in which several jobs of few different types are produced periodically.

Assigning jobs to time frames on a single machine to minimize total tardiness

This paper deals with a sequencing problem arising in the management of paced-flowlines, that is production lines where jobs are released at constant time intervals. The problem is to sequence jobs to minimize total tardiness. The problem can be formulated as an assignment problem with a number of knapsack constraints. We prove the strong NP-hardness of the problem and give a number of lower bounds which are used in a branch-and-bound algorithm. Computational results in realistic settings confirm the effectiveness of the procedure developed. The results are particularly interesting with reference to mixed-model assembly lines in which several jobs of few different types are produced periodically.

New polynomial bounds for matroidal knapsacks

The Matroidal Knapsack Problem (MK) consists in finding the maximum weight basis for a given matroid subject to a knapsack constraint. Special cases of this problem are the Multiple Choice Knapsack and the Capacitated Minimal Spanning Tree Problems. Using matroidal relaxations for knapsack problems we build a relaxation for MK. This relaxation produces an upper bound on MK dominating the usual LP-bound and computable using a polynomial number of calls to the greedy algorithm for the matroid.

On using clique overlapping for detecting knapsack constraint redundancy and infeasibility in 0–1 mixed integer programs

In this note we present fast procedures for detecting knapsack constraint redundancy and infeasibility in 0–1 mixed integer programs by using information from probing analysis and overlapping clique identification. The new procedures improve current preprocessing techniques for size reduction of integer programs.

On Prize‐collecting Tours and the Asymmetric Travelling Salesman Problem

We consider a variant of the Travelling Salesman Problem which is to determine a tour visiting each vertex in the graph at most at one time; if a vertex is left unrouted a given penalty has to be paid. The objective function is to find a balance between these penalties and the cost of the tour. We call this problem the Profitable Tour Problem(PTP). If, in addition, each vertex is associated with a prize and there is a knapsack constraint which guarantees that a sufficiently large prize is collected, we have the well‐known Prize‐collecting Travelling Salesman Problem (PCTSP). In this paper we summarize the main results presented in the literature, then we give lower bounds for the asymmetric version of PTP and PCTSP. Moreover, we show, through computational experiments, that large size instances of the asymmetric PTP can be solved exactly.

On prize-collecting tours and the asymmetric travelling salesman problem

We consider a variant of the Travelling Salesman Problem which is to determine a tour visiting each vertex in the graph at most at one time; if a vertex is left unrouted a given penalty has to be paid. The objective function is to find a balance between these penalities and the cost of the tour. We call this problem the Profitable Tour Problem (PTP). If, in addition, each vertex is associated with a prize and there is a knapsack constraint which guarantees that a sufficiently large prize is collected, we have the well-known Prize-collecting Travelling Salesman Problem (PCTSP). In this paper we summarize the main results presented in the literature, then we give lower bounds for the asymmetric version of PTP and PCTSP. Moreover, we show, through computational experiments, that large size instances of the asymmetric PTP can be solved exactly.

A correction of the justification of the Dietrich-escudero-garín-pérez O(n) procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates

In this brief note we present a correction of the Propositions 1 and 5 for proving the O(n) complexity of the algorithms described in [Dietrich et al. (1993)] for identifying maximal cliques and non-dominated covers from extensions of consecutive minimal covers and alternates implied by 0–1 knapsack constraints.

Obtaining clique, cover and coefficient reduction inequalities as Chvatal-Gomory inequalities and Gomory fractional cuts

We show how clique and cover induced inequalities implied from 0–1 knapsack constraints can be obtained as Chvatal-Gomory inequalities. LP tighter and 0–1 equivalent constraints to 0–1 knapsack constraints obtained by the ‘big M’ reduction procedure can also be generated as Chvatal-Gomory inequalities. We also show how some extended coefficient reduction based LP tighter and 0–1 equivalent constraints can be generated as Gomory fractional cuts.

Partial cover and complete cover inequalities

In this paper we use an extension of the well-known cover inequalities to obtain valid inequalities for 0–1 integer problems with two different sided knapsack constraints and, in general, for any kind of 0–1 integer problems having at least two different sided constraints. These inequalities cannot be derived from any of the individual constraints alone and are stronger than the cover inequalities derived from knapsack constraints individually. Given a solution, we state the conditions under which violated inequalities of this type exist, and we also propose heuristics to identify them. The application of these inequalities to different classes of 0–1 integer problems is also studied.

O(n) Procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates

In this paper, we describe computationally efficient procedures for identifying all maximal cliques and non-dominated selected subsets of extensions of minimal covers and alternates that are implied by single 0–1 knapsack constraints. The induced inequalities are satisfied by and 0–1 feasible solution to the knapsack constraint, but are tipically violated by fractional solutions. In addition, the procedures described here are used in conjunction with other constraints to further tighten LP relaxations of 0–1 programs. The complexity of the procedures isO(n).

Efficient reformulation for 0–1 programs — methods and computational results

We introduce two general methods for 0–1 program reformulation. Our first method generalizes coefficient reduction, our second method generalizes lifting. Together they provide a unifying interpretation of many previously described automatic reformulation methods. The particular model structures that we consider are individual knapsack constraints, pairs of knapsack constraints, clique and cover induced inequalities, variable upper bounding constraints and capacity expansion constraints. We describe several easy applications of our reformulation procedures. Some computational experience is reported, including the currently best known results on a well-known 147 × 2655 benchmark problem.