Knapsack Cover Research Articles

Lifting for the integer knapsack cover polyhedron

Lifting for the integer knapsack cover polyhedron

Scattered storage assignment: Mathematical model and valid inequalities to optimize the intra-order item distances

In scattered storage, individual items are intentionally distributed across multiple positions in the picking area. Especially in e-commerce environments, where orders typically consist of a few items in small quantities, such a storage policy can reduce picking travel times by increasing the likelihood that items belonging to the same order can be found in nearby positions. In this paper, we propose a scattered storage policy that, when determining the position where each replenished item should be stored, tries to minimize the sum of pairwise distances (SPD) between all items belonging to the same order including a drop-off point. However, when solving the exact mathematical model using integer programming techniques (0–1), the combinatorial nature of the problem hinders performance. We show that the solutions of the MIP solver can be improved by relaxing some variables and adding valid inequalities based on graph properties and knapsack cover constraints. Finally, we prove that the SPD objective is 65% lower for our scattered storage policy than for a traditional volume based storage policy.

Directed fixed charge multicommodity network design: A cutting plane approach using polar duality

We present an efficient cutting-plane based approach to exactly solve a directed fixed charge network design (DFCND) problem, wherein the valid inequalities to the problem are generated using the polar duality approach. The biggest challenge in using this approach arises in constructing the polar dual of the problem. This would require enumerating all the extreme points of the convex hull of DFCND, which is computationally impractical for any instance of a reasonable size. Moreover, the resulting polar dual would be too large to solve efficiently, which is required at every iteration of the cutting-plane algorithm. The novelty of our solution approach lies in suggesting a way to circumvent this challenge by instead generating the violated facets, using polar duality, of the smaller substructures involving only a small subset of constraints and variables, obtained from 2-, 3-and 4-partitions of the underlying graph. For problem instances based on sparse graphs with zero flow costs, addition of these inequalities closes more than 20% of the optimality gap remaining after the addition of the knapsack cover inequalities used in the literature. This allows us to solve the problem instances in less than 400 s, on average, which otherwise take around 1000 s with the addition of only the knapsack cover inequalities, and around 4 hours for the Cplex MIP solver at its default setting.

Lifting the knapsack cover inequalities for the knapsack polytope

Valid inequalities for the knapsack polytope have proven to be very useful in exact algorithms for mixed-integer linear programming. In this paper, we focus on the knapsack cover inequalities, introduced in 2000 by Carr and co-authors. In general, these inequalities can be rather weak. To strengthen them, we use lifting. Since exact lifting can be time-consuming, we present two fast approximate lifting procedures. The first procedure is based on mixed-integer rounding, whereas the second uses superadditivity.

Approximate and exact merging of knapsack constraints with cover inequalities

ABSTRACT This paper presents both approximate and exact merged knapsack cover inequalities, a class of cutting planes for knapsack and multiple knapsack integer programs. These inequalities combine the information from knapsack constraints and cover inequalities. Approximate merged knapsack cover inequalities can be generated through a algorithm, where n is the number of variables. This class of inequalities can be strengthened to an exact version with a pseudo-polynomial time algorithm. Computational experiments demonstrate an average improvement of approximately in solution time and in the number of ticks from CPLEX when approximate merged knapsack cover inequalities are implemented as preprocessing cuts to solve some benchmark multiple knapsack problems. Furthermore, exact merged knapsack cover inequalities improve the solution time and number of ticks of some random multiple knapsack instances by and , respectively.

The incremental connected facility location problem

We consider the incremental connected facility location problem (incremental ConFL), in which we are given a set of potential facilities, a set of interconnection nodes, a set of customers with demands, and a planning horizon. For each time period, we have to select a set of facilities to open, a set of customers to be served, the assignment of these customers to the open facilities, and a network that connects the open facilities. Once a customer is served, it must remain served in subsequent periods. Furthermore, in each time period the total demand of all customers served must be at least equal to a given minimum coverage requirement for that period. The objective is to minimize the total cost for building the network given by the investment and maintenance costs for the facilities and the network summed up over all time periods.We propose a mixed integer programming approach in which, in each time period, a single period ConFL with coverage restrictions has to be solved. For this latter problem, which is of particular interest in itself, new families of valid inequalities are proposed: these are set union knapsack cover (SUKC) inequalities, which are further enhanced by lifting and/or combined with cut-set inequalities, which are primarily used to ensure connectivity requirements. Details of an efficient branch-and-cut implementation are presented and computational results on a benchmark set of large instances are given, including examples of telecommunication networks in Germany.

A Branch-and-Price-and-Cut Algorithm for Resource-Constrained Pickup and Delivery Problems

We study a multicommodity, multiperiod, resource-constrained pickup-and-delivery problem inspired by the short-term planning of maintenance services at offshore wind farms. To begin a maintenance service, different types of relatively scarce servicemen need to be delivered (transported) to the service locations. We develop resource-exceeding route (RER) inequalities, which are inspired by knapsack cover inequalities, to model the scarcity of servicemen. In addition to a traditional separation approach, we present a column-dependent constraints approach so as to include the RER inequalities in the mathematical formulation. An alternative pricing strategy is developed to correctly include the column-dependent constraints. The resulting approach is broadly applicable to any routing problem that involves a set of scarce resources. We present a branch-and-price-and-cut algorithm to compare both approaches that include RER inequalities. The branch-and-price-and-cut algorithm relies on efficiently solving a new variant of the elementary resource-constrained shortest-path problem, using a tailored pulse algorithm developed specifically to solve it. Computational experiments show that the RER inequalities significantly tighten the root node relaxations. The column-dependent constraints approach then searches the branch-and-bound tree more effectively and appears to be competitive with the traditional separation procedure. Both approaches are able to solve instances of up to 92 nodes over 21 periods to optimality.

Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any $\varepsilon >0$, there is a $(1/\varepsilon)^{O(1)}n^{O(\log n)}$-size LP relaxation with an integrality gap of at most $2+\varepsilon$, where $n$ is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on $O(\log^2 n)$-depth monotone circuits with fan-in~$2$ for evaluating weighted threshold functions with $n$ inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.

An effective dynamic programming algorithm for the minimum-cost maximal knapsack packing problem

Given a set of items with profits and weights and a knapsack capacity, we study the problem of finding a maximal knapsack packing that minimizes the profit of the selected items. We propose an effective dynamic programming (DP) algorithm which has a pseudo-polynomial time complexity. We demonstrate the equivalence between this problem and the problem of finding a minimal knapsack cover that maximizes the profit of the selected items. In an extensive computational study on a large and diverse set of benchmark instances, we demonstrate that the new DP algorithm outperforms a state-of-the-art commercial mixed-integer programming (MIP) solver applied to the two best performing MIP models from the literature.

A polyhedral study on chance constrained program with random right-hand side

The essential structure of the mixed-integer programming formulation for chance-constrained program (CCP) is the intersection of multiple mixing sets with a 0–1 knapsack. To improve our computational capacity on CCP, an underlying substructure, the (single) mixing set with a 0–1 knapsack, has received substantial attentions recently. In this study, we consider a CCP problem with stochastic right-hand side under a finite discrete distribution. We first present a family of strong inequalities that subsumes known facet-defining ones for that single mixing set. Due to the flexibility of our generalized inequalities, we develop a new separation heuristic that has a complexity much less than existing one and guarantees generated cutting planes are facet-defining for the polyhedron of CCP. Then, we study lifting and superadditive lifting on knapsack cover inequalities, and provide an implementable procedure on deriving another family of strong inequalities for the single mixing set. Finally, different from the traditional approach that aggregates original constraints to investigate polyhedral implications due to their interactions, we propose a novel blending procedure that produces strong valid inequalities for CCP by integrating those derived from individual mixing sets. We show that, under certain conditions, they are the first type of facet-defining inequalities describing intersection of multiple mixing sets, and design an efficient separation heuristic for implementation. In the computational experiments, we perform a systematic study and illustrate the efficiency of the proposed inequalities on solving chance constrained static probabilistic lot-sizing problems.

Primal–Dual Algorithms for Precedence Constrained Covering Problems

A covering problem is an integer linear program of type $$\min \{c^Tx\mid Ax\ge D,\ 0\le x\le d,\ x \in \mathbb {Z}\}$$min{cTxźAxźD,0≤x≤d,xźZ} where $$A\in \mathbb {Z}^{m\times n}_+$$AźZ+m×n, $$D\in \mathbb {Z}_+^m$$DźZ+m, and $$c,d\in \mathbb {Z}_+^n$$c,dźZ+n. In this paper, we study covering problems with additional precedence constraints $$\{x_i\le x_j \ \forall j\preceq i \in \mathcal {P}\}$${xi≤xjźjźiźP}, where $$\mathcal {P}=([n], \preceq )$$P=([n],ź) is some arbitrary, but fixed partial order on the items represented by the column-indices of A. Such precedence constrained covering problems (PCCPs) are of high theoretical and practical importance even in the special case of the precedence constrained knapsack problem, that is, where $$m=1$$m=1 and $$d\equiv 1$$dź1. Our main result is a strongly-polynomial primal---dual approximation algorithm for PCCP with $$d\equiv 1$$dź1. Our approach generalizes the well-known knapsack cover inequalities to obtain an IP formulation which renders any explicit precedence constraints redundant. The approximation ratio of this algorithm is upper bounded by the width of $$\mathcal {P}$$P, that is, by the size of a maximum antichain in $$\mathcal {P}$$P. Interestingly, this bound is independent of the number of constraints. We are not aware of any other results on approximation algorithms for PCCP on arbitrary posets $$\mathcal {P}$$P. For the general case with $$d\not \equiv 1$$dź1, we present pseudo-polynomial algorithms. Finally, we show that the problem does not admit a PTAS under standard complexity assumptions.

Lifting for mixed integer programs with variable upper bounds

We investigate the convex hull of the set defined by a single inequality with continuous and binary variables, which are additionally related by variable upper bound constraints. First we elaborate on general sequence dependent lifting for this set and present a dynamic program for calculating lifting coefficients. Then we study variable fixings of this set to knapsack covers and to the single binary variable polytope. We explicitly give lifting coefficients of continuous variables when lifting the knapsack cover inequality. We provide two new families of facet-defining inequalities for the single binary variable polytope and we prove that combined with the trivial inequalities they give a full description of this polytope.

Approximation algorithms for the partition vertex cover problem

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G=(V,E), a cost function c:V→Z+, a partition P1,…,Pr of the edge set E, and a parameter ki for each partition Pi. The objective is to find a minimum cost set of vertices which cover at least ki edges from the partition Pi. We call this the Partition-VC problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O(log⁡r), where r is the number of sets in the partition of the edge set. We also extend our result to more general settings. For example we consider a problem where additionally edges have profits, and we would like to pick a minimum cost set of vertices which cover edges of total profit at least Πi for each partition Pi. We call this the Knapsack Partition Vertex Cover problem. We give an O(log⁡r) approximation algorithm for this problem as well.

Approximability of Capacitated Network Design

In the capacitated survivable network design problem (Cap-SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a o(m) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP.We give an O(logn) approximation for a special case of Cap-SNDP where the global minimum cut is required to be at least R. (Note that this problem generalizes the min-cost λ-edge-connected subgraph problem, which is the special case of our problem when all capacities are unit.) Our result is based on a rounding of a natural cut-based LP relaxation strengthened with knapsack-cover (KC) inequalities. Our technique extends to give a similar approximation for a new network design problem that captures global minimum cut as a special case. We then show that as we move away from global connectivity, even for the single pair case (that is, when only one pair (s,t) has positive connectivity requirement), this strengthened LP has Ω(n) integrality gap.We also consider a variant of Cap-SNDP in which multiple copies of an edge can be bought: we give an O(logk) approximation for this case, where k is the number of vertex pairs with non-zero connectivity requirement. This improves upon the previously known O(min{k,logR max})-approximation when R max is large; here R max is the largest requirement. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω(loglogn)-hard to approximate even for the single-source version of the problem.

A mixed integer linear programming approach to minimize the number of late jobs with and without machine availability constraints

This study investigates scheduling problems that occur when the weighted number of late jobs that are subject to deterministic machine availability constraints have to be minimized. These problems can be modeled as a more general job selection problem. Cases with resumable, non-resumable, and semi-resumable jobs as well as cases without availability constraints are investigated. The proposed efficient mixed integer linear programming approach includes possible improvements to the model, notably specialized lifted knapsack cover cuts. The method proves to be competitive compared with existing dedicated methods: numerical experiments on randomly generated instances show that all 350-job instances of the test bed are closed for the well-known problem 1|ri|∑wiUi. For all investigated problem types, 98.4% of 500-job instances can be solved to optimality within 1hour.

Randomized Competitive Algorithms for Generalized Caching

We consider online algorithms for the generalized caching problem. Here we are given a cache of size k and pages with arbitrary sizes and fetching costs. Given a request sequence of pages, the goal is to minimize the total cost of fetching the pages into the cache. Our main result is an online algorithm with competitive ratio $O(\log^2k)$, which gives the first $o(k)$ competitive algorithm for the problem. We also give improved $O(\log k)$-competitive algorithms for the special cases of the bit model and fault model, improving upon the previous $O(\log^2k)$ guarantees due to Irani [Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997, pp. 701–710]. Our algorithms are based on an extension of the online primal-dual framework introduced by Buchbinder and Naor [Math. Oper. Res., 34 (2009), pp. 270–286] and involve two steps. First, we obtain an $O(\log k)$-competitive fractional algorithm based on solving online an LP formulation strengthened with exponentially many knapsack cover constraints. Second, we design a suitable online rounding procedure to convert this online fractional algorithm into a randomized algorithm. Our techniques provide a unified framework for caching algorithms and are substantially simpler than those previously used.

Practical strategies for generating rank-1 split cuts in mixed-integer linear programming

In this paper we propose practical strategies for generating split cuts, by considering integer linear combinations of the rows of the optimal simplex tableau, and deriving the corresponding Gomory mixed-integer cuts; potentially, we can generate a huge number of cuts. A key idea is to select subsets of variables, and cut deeply in the space of these variables. We show that variables with small reduced cost are good candidates for this purpose, yielding cuts that close a larger integrality gap. An extensive computational evaluation of these cuts points to the following two conclusions. The first is that our rank-1 cuts improve significantly on existing split cut generators (Gomory cuts from single tableau rows, MIR, Reduce-and-Split, Lift-and-Project, Flow and Knapsack cover): on MIPLIB instances, these generators close 24% of the integrality gap on average; adding our cuts yields an additional 5%. The second conclusion is that, when incorporated in a Branch-and-Cut framework, these new cuts can improve computing time on difficult instances.

Clique-based facets for the precedence constrained knapsack problem

We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP polyhedron based on clique inequalities. A comparison with existing techniques, that lift knapsack cover inequalities for the PCKP, is also presented. It is shown that the clique-based approach generates facets that cannot be found through the existing cover-based approaches, and that the addition of clique-based inequalities for the PCKP can be computationally beneficial, for both PCKP instances arising in real applications, and applications in which PCKP appears as an embedded structure.

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.

Higher-order cover cuts from zero–one knapsack constraints augmented by two-sided bounding inequalities

Extending our work on second-order cover cuts [F. Glover, H.D. Sherali, Second-order cover cuts, Mathematical Programming (ISSN: 0025-5610 1436-4646) (2007), doi:10.1007/s10107-007-0098-4. (Online)], we introduce a new class of higher-order cover cuts that are derived from the implications of a knapsack constraint in concert with supplementary two-sided inequalities that bound the sums of sets of variables. The new cuts can be appreciably stronger than the second-order cuts, which in turn dominate the classical knapsack cover inequalities. The process of generating these cuts makes it possible to sequentially utilize the second-order cuts by embedding them in systems that define the inequalities from which the higher-order cover cuts are derived. We characterize properties of these cuts, design specialized procedures to generate them, and establish associated dominance relationships. These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation. We also discuss a lifting procedure for further tightening any generated cut, and establish its polynomial-time operation for unit-coefficient cuts. A numerical example is presented that illustrates these procedures and the relative strength of the generated non-redundant, non-dominated higher-order cuts, all of which turn out to be facet-defining for this example. Some preliminary computational results are also presented to demonstrate the efficacy of these cuts in comparison with lifted minimal cover inequalities for the underlying knapsack polytope.