Improving problem reduction for 0-1 Multidimensional Knapsack Problems with valid inequalities

© 2016 Elsevier Ltd. All rights reserved. This paper investigates the problem reduction heuristic for the Multidimensional Knapsack Problem (MKP). The MKP formulation is first strengthened by the Global Lifted Cover Inequalities (GLCI) using the cutting plane approach. The dynamic core problem heuristic is then applied to find good solutions. The GLCI is described in the general lifting framework and several variants are introduced. A Two-level Core problem Heuristic is also proposed to tackle large instances. Computational experiments were carried out on classic benchmark problems to demonstrate the effectiveness of this new method

A Combinatorial Optimization Approach to the Selection of Statistical Units

In the case of some large statistical surveys, the set of units that will constitute the scope of the survey must be selected. We focus on the real case of a Census of Agriculture, where the units are farms. Surveying each unit has a cost and brings a different portion of the whole information. In this case, one wants to determine a subset of units producing the minimum total cost for being surveyed and representing at least a certain portion of the total information. Uncertainty aspects also occur, because the portion of information corresponding to each unit is not perfectly known before surveying it. The proposed approach is based on combinatorial optimization, and the arising decision problems are modeled as multidimensional binary knapsack problems. Experimental results show the effectiveness of the proposed approach

Cutting Plane Algorithms for 0-1 Programming Based on Cardinality Cuts

Cataloged from PDF version of article.We present new valid inequalities for 0-1 programming problems that work in similar ways to well known cover inequalities. Discussion and analysis of these cuts is followed by their revision and use in integer programming as a new generation of cuts that excludes not only portions of polyhedra containing noninteger points, also parts with some integer points that have been explored in search of an optimal solution. Our computational experimentations demonstrate that this new approach has significant potential for solving large scale integer programming problems. 2010 Elsevier B.V. All rights reserved

Knapsack Problems with Side Constraints

The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once. The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem. These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts

An Expandable Machine Learning-Optimization Framework to Sequential Decision-Making

We present an integrated prediction-optimization (PredOpt) framework to efficiently solve sequential decision-making problems by predicting the values of binary decision variables in an optimal solution. We address the key issues of sequential dependence, infeasibility, and generalization in machine learning (ML) to make predictions for optimal solutions to combinatorial problems. The sequential nature of the combinatorial optimization problems considered is captured with recurrent neural networks and a sliding-attention window. We integrate an attention-based encoder-decoder neural network architecture with an infeasibility-elimination and generalization framework to learn high-quality feasible solutions to time-dependent optimization problems. In this framework, the required level of predictions is optimized to eliminate the infeasibility of the ML predictions. These predictions are then fixed in mixed-integer programming (MIP) problems to solve them quickly with the aid of a commercial solver. We demonstrate our approach to tackling the two well-known dynamic NP-Hard optimization problems: multi-item capacitated lot-sizing (MCLSP) and multi-dimensional knapsack (MSMK). Our results show that models trained on shorter and smaller-dimensional instances can be successfully used to predict longer and larger-dimensional problems. The solution time can be reduced by three orders of magnitude with an average optimality gap below 0.1%. We compare PredOpt with various specially designed heuristics and show that our framework outperforms them. PredOpt can be advantageous for solving dynamic MIP problems that need to be solved instantly and repetitively

Applications of combinatorial optimization arising from large scale surveys

Many difficult statistical problems arising in censuses or in other large scale surveys have an underlying Combinatorial Optimization structure and can be solved with Combinatorial Optimization techniques. These techniques are often more efficient than the ad hoc solution techniques already developed in the field of Statistics. This thesis considers in detail two relevant cases of such statistical problems, and proposes solution approaches based on Combinatorial Optimization and Graph Theory. The first problem is the delineation of Functional Regions, the second one concerns the selection of the scope of a large survey, as briefly described below. The purpose of this work is therefore the innovative application of known techniques to very important and economically relevant practical problems that the "Censuses, Administrative and Statistical Registers Department" (DICA) of the Italian National Institute of Statistics (Istat), where I am senior researcher, has been dealing with. In several economical, statistical and geographical applications, a territory must be partitioned into Functional Regions. This operation is called Functional Regionalization. Functional Regions are areas that typically exceed administrative boundaries, and they are of interest for the evaluation of the social and economical phenomena under analysis. Functional Regions are not fixed and politically delimited, but are determined only by the interactions among all the localities of a territory. In this thesis, we focus on interactions represented by the daily journey-to-work flows between localities in which people live and/or work. Functional Regionalization of a territory often turns out to be computationally difficult, because of the size (that is, the number of localities constituting the territory under study) and the nature of the journey-to-work matrix (that is, the sparsity). In this thesis, we propose an innovative approach to Functional Regionalization based on the solution of graph partition problems over an undirected graph called transitions graph, which is generated by using the journey-to-work data. In this approach, the problem is solved by recursively partitioning the transition graph by using the min cut algorithms proposed by Stoer and Wagner and Brinkmeier. %In the second approach, the problem is solved maximizing a function of the sizes and interactions of subsets identified by successions of partitions obtained via Multilevel partitioning approach. This approach is applied to the determination of the Functional Regions for the Italian administrative regions. The target population of a statistical survey, also called scope, is the set of statistical units that should be surveyed. In the case of some large surveys or censuses, the scope cannot be the set of all available units, but it must be selected from this set. Surveying each unit has a cost and brings a different portion of the whole information. In this thesis, we focus on the case of Agricultural Census. In this case, the units are farms, and we want to determine a subset of units producing the minimum total cost and safeguarding at least a certain portion of the total information, according to the coverage levels assigned by the European regulations. Uncertainty aspects also occur, because the portion of information corresponding to each unit is not perfectly known before surveying it. The basic decision aspect is to establish the inclusion criteria before surveying each unit. We propose here to solve the described problem using multidimensional binary knapsack models

Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17/9 + \epsilon) < 1.89 approximation, which improves to (558/325 + \epsilon) < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. As a second major and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also, in this case, the best-known polynomial-time approximation factor (even for the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3/2 + \epsilon)-approximation for 2DKR, which improves to (4/3 + \epsilon) in the cardinality case.Comment: 64pages, full version of FOCS 2017 pape

Applications of combinatorial optimization arising from large scale surveys

Many difficult statistical problems arising in censuses or in other large scale surveys have an underlying Combinatorial Optimization structure and can be solved with Combinatorial Optimization techniques. These techniques are often more efficient than the ad hoc solution techniques already developed in the field of Statistics. This thesis considers in detail two relevant cases of such statistical problems, and proposes solution approaches based on Combinatorial Optimization and Graph Theory. The first problem is the delineation of Functional Regions, the second one concerns the selection of the scope of a large survey, as briefly described below. The purpose of this work is therefore the innovative application of known techniques to very important and economically relevant practical problems that the "Censuses, Administrative and Statistical Registers Department" (DICA) of the Italian National Institute of Statistics (Istat), where I am senior researcher, has been dealing with. In several economical, statistical and geographical applications, a territory must be partitioned into Functional Regions. This operation is called Functional Regionalization. Functional Regions are areas that typically exceed administrative boundaries, and they are of interest for the evaluation of the social and economical phenomena under analysis. Functional Regions are not fixed and politically delimited, but are determined only by the interactions among all the localities of a territory. In this thesis, we focus on interactions represented by the daily journey-to-work flows between localities in which people live and/or work. Functional Regionalization of a territory often turns out to be computationally difficult, because of the size (that is, the number of localities constituting the territory under study) and the nature of the journey-to-work matrix (that is, the sparsity). In this thesis, we propose an innovative approach to Functional Regionalization based on the solution of graph partition problems over an undirected graph called transitions graph, which is generated by using the journey-to-work data. In this approach, the problem is solved by recursively partitioning the transition graph by using the min cut algorithms proposed by Stoer and Wagner and Brinkmeier. %In the second approach, the problem is solved maximizing a function of the sizes and interactions of subsets identified by successions of partitions obtained via Multilevel partitioning approach. This approach is applied to the determination of the Functional Regions for the Italian administrative regions. The target population of a statistical survey, also called scope, is the set of statistical units that should be surveyed. In the case of some large surveys or censuses, the scope cannot be the set of all available units, but it must be selected from this set. Surveying each unit has a cost and brings a different portion of the whole information. In this thesis, we focus on the case of Agricultural Census. In this case, the units are farms, and we want to determine a subset of units producing the minimum total cost and safeguarding at least a certain portion of the total information, according to the coverage levels assigned by the European regulations. Uncertainty aspects also occur, because the portion of information corresponding to each unit is not perfectly known before surveying it. The basic decision aspect is to establish the inclusion criteria before surveying each unit. We propose here to solve the described problem using multidimensional binary knapsack models