Set Packing Research Articles

Nonprofit peer-to-peer ridesharing optimization

Both for-profit and nonprofit peer-to-peer (P2P) ridesharing services have gained enormous popularity in recent years due to their advantages over solo driving and public transit. We study the rideshare matching and routing problem in a nonprofit P2P ridesharing system consisting of a matching agency, drivers and riders. The matching agency is a government or a not-for-profit organization and its objective is to maximize the societal benefits of ridesharing. The drivers involved are commuters and hence have their own travel plans, which are executed regardless of whether riders are matched with them. We consider both static and dynamic versions of the nonprofit P2P ridesharing problem. Existing modeling and solution approaches for similar P2P ridesharing problems can only solve relatively small problem instances optimally. We propose an exact solution algorithm for the static version of the problem by taking advantage of its special characteristics. This exact solution approach formulates and solves the problem as a set packing formulation using route-based variables, and uses an efficient graph-based approach to generate all necessary vehicle routes in the formulation quickly. We also develop a column generation (CG) based heuristic approach for the static problem. Finally, we propose two dynamic dispatching policies for the dynamic version of the problem. Our proposed exact algorithm solves very large problem instances (e.g., with 600 drivers and 1800 riders) of the static problem and our CG based heuristic can find near-optimal solutions for even larger instances of the static problem in short computation time. Our dynamic dispatching policies can generate near-optimal solutions for the dynamic problem in real-time fashion. We also generate some important insights based on some taxi trip data in Washington DC. First, P2P ridesharing can bring significant cost-saving, especially when the participants have a relatively flexible schedule. For every 10% increase in schedule flexibility, there is an about 4% to 7% increase in cost-saving. Second, the cost-saving due to ridesharing increases with the vehicle capacity, but this increase slows down quickly when the vehicle capacity reaches 4. Third, ridesharing generates more cost savings during peak hours and in urban areas.

The double-assignment plant location problem with co-location

In this paper, a new variant of the Simple Plant Location Problem is proposed. We consider additional conditions in the classic location-allocation problem for clients and facilities. Namely, some pairs have to be served by a common plant. The resulting problem can be addressed with existing models for the case of single assignment. However, to the best of our knowledge, the proposed setting when each client must be assigned to a couple of facilities is still unexplored. We examine the implications of adding such new constraints to standard formulations of the SPLP with double assignment. We compare the resulting formulations from a theoretical point of view. After that, we focus on the study of one of the models, which turned out to be a set packing problem. All the clique facets are identified and a separation algorithm is devised. Although the separation problem is proved to be NP-hard, our computational experience shows that the separation algorithm is effective and efficient, reducing computational times and duality gaps for all the instances tested.

Data Association via Set Packing for Computer Vision Applications

Significant progress has been made in the field of computer vision because of the development of supervised machine learning algorithms, which efficiently extract information from high-dimensional data such as images and videos. Such techniques are particularly effective at recognizing the presence or absence of entities in the domains where labeled data are abundant. However, supervised learning is not sufficient in applications where one needs to annotate each unique entity in crowded scenes respecting known domain-specific structures of those entities. This problem, known as data association, provides fertile ground for the application of combinatorial optimization. In this review paper, we present a unified framework based on column generation for some computer vision applications, namely multiperson tracking, multiperson pose estimation, and multicell segmentation, which can be formulated as set packing problems with a massive number of variables. To solve them, column generation algorithms are applied to circumvent the need to enumerate all variables explicitly. To enhance the solution process, we provide a general approach for applying subset-row inequalities to tighten the formulations and introduce novel dual-optimal inequalities to reduce the dual search space. The proposed algorithms and their enhancements are successfully applied to solve the three aforementioned computer vision problems and achieve superior performance over benchmark approaches. The common framework presented allows us to leverage operations research methodologies to efficiently tackle computer vision problems.

A hybrid adaptive large neighborhood search heuristic for the team orienteering problem

The Team Orienteering Problem (TOP) is a well-known NP-Hard vehicle routing problem in which one maximizes the collected profits for visiting some nodes. In this paper, we propose a Hybrid Adaptive Large Neighborhood Search (HALNS) to solve this problem. Our algorithm combines the exploration power of ALNS with local search procedures and an optimization stage using a Set Packing Problem to further improve the solutions. Extensive computational experiments demonstrate the high performance of our HALNS outperforming all the competing algorithms in the literature on a large set of benchmark instances in terms of solution quality and/or computational time. Our HALNS identifies all the 387 Best Known Solutions (BKS) from the literature on a first dataset including small-scale benchmark instances and all the 333 BKS for large-scale benchmark instances within very short computational times. Moreover, we improve one large-scale instance solution.

A distributed algorithm for a maximal 2-packing set in Halin graphs

In this work, we propose Maximal-2-Packing-Halin, a distributed algorithm that finds a maximal 2-packing set in undirected non-geometric Halin graphs of order n in linear time. First, this algorithm finds an external face of the input graph through the application of graph-reduction rules. Second, each vertex determines if it belongs to a maximal 2-packing set by applying a set of vertex-coloring rules.

Accelerating Column Generation via Flexible Dual Optimal Inequalities with Application to Entity Resolution

In this paper, we introduce a new optimization approach to Entity Resolution. Traditional approaches tackle entity resolution with hierarchical clustering, which does not benefit from a formal optimization formulation. In contrast, we model entity resolution as correlation-clustering, which we treat as a weighted set-packing problem and write as an integer linear program (ILP). In this case, sources in the input data correspond to elements and entities in output data correspond to sets/clusters. We tackle optimization of weighted set packing by relaxing integrality in our ILP formulation. The set of potential sets/clusters can not be explicitly enumerated, thus motivating optimization via column generation. In addition to the novel formulation, we also introduce new dual optimal inequalities (DOI), that we call flexible dual optimal inequalities, which tightly lower-bound dual variables during optimization and accelerate column generation. We apply our formulation to entity resolution (also called de-duplication of records), and achieve state-of-the-art accuracy on two popular benchmark datasets. Our F-DOI can be extended to other weighted set-packing problems.

Packing, partitioning, and covering symresacks

In this paper, we consider symmetric binary programs that contain set packing, partitioning, or covering inequalities. To handle symmetries as well as set packing, partitioning, or covering constraints simultaneously, we introduce constrained symresacks which are the convex hulls of all binary points that are lexicographically not smaller than their image w.r.t. a coordinate permutation and which fulfill packing, partitioning, or covering constraints. We show that linear optimization problems over constrained symresacks can be solved in cubic time. Furthermore, we derive complete linear descriptions of constrained symresacks for particular classes of symmetries. These inequalities can then be used as strong symmetry handling cutting planes in a branch-and-bound procedure. Numerical experiments show that we can benefit from incorporating set packing, partitioning, or covering constraints into symmetry handling inequalities.

Risk-informed reliability improvement optimization for verification and validation planning based on set covering modeling

Verification and validation plays a crucial role in the new product development process since it is able to assure product performance and eventually determine customer satisfaction. The verification and validation planning assigns a set of verification and validation activities such as computational simulations and physical tests, which are proposed to mitigate the design risks of specific failure modes, to verify and validate that the product conforms to the design objectives. This article formulates the optimal verification and validation planning using set covering, set partition, and set packing concepts. An extended optimization model based on set covering is also formulated, which can well accommodate the requirements including minimizing the overall risk of the product design, meeting pre-specified risk thresholds of specific failure modes, and covering identified critical failure modes. Additional constraints such as the implementation sequences, time gaps of various types of verification and validation activities, and the distinct effectiveness of each verification and validation activity in reducing design risks are considered in the verification and validation planning optimization models. The decay of the improvement effectiveness of a failure mode with multiple verification and validation activities over time is also considered. The application of the proposed mathematical optimization models for product verification and validation planning is illustrated through the product development of a power generation system within a diesel engine.

A genetic algorithm for the maximum 2-packing set problem

A genetic algorithm for the maximum 2-packing set problem

A framework to exploit the structure of and solve set packing problems with a semi-block-angular structure

Real-world, large-scale problems often have coefficient matrices with special structures that can be obtained by reordering their rows and columns. One fundamental problem in the field of combinatorial optimization is the Set Packing Problem (SPP), which takes on added significance in view of its far-reaching applications in routing and scheduling problems, scheduling surgical procedures, the winner determination problem, forestry, production management, etc. The coefficient matrix of this problem has low density (the number of 1’s is much less than the number of 0’s in the matrix); therefore, by reordering the matrix, one can aggregate its 1’s in order to exploit special structures like semi-block-angular ones.In this paper, a decomposition technique based on constraint partitioning is proposed for exploiting the semi-block-angular structures of SPPs and solving the original problem through solving the sub-problems of the obtained structure. In addition, a theorem is developed to relate the optimal values of the original problem to its sub-problems.The results of implementing the proposed algorithm on some sample test problems demonstrate the ability of the algorithm to exploit semi-block-angular structures and achieve optimal solutions for SPPs. In addition, the application of the proposed algorithm to solve SPPs without a semi-block-angular structure is also investigated on some test problems of OR-library.

The quality of equilibria for set packing and throughput scheduling games

We introduce set packing games as an abstraction of situations in which n selfish players select disjoint subsets of a finite set of indivisible items, and analyze the quality of several equilibria for this basic class of games. Special attention is given to a subclass of set packing games, namely throughput scheduling games, where the items represent jobs, and the subsets that a player can select are those jobs that this player can schedule feasibly. We show that the quality of three types of equilibrium solutions is only moderately suboptimal. Specifically, the paper gives tight bounds on the price of anarchy for Nash equilibria, subgame perfect equilibria of games with sequential play, and k-collusion Nash equilibria. Under the assumption that players are allowed to play suboptimally and achieve an alpha -approximate equilibrium, our tight price of anarchy bounds are alpha +1 for Nash and subgame perfect equilibria, but less than alpha +1/(e-1) for subgame perfect equilibria when games are symmetric. For k-collusion Nash equilibria, the price of anarchy equals alpha +(n-k)/(n-1), where 1le kle n.

An evolutionary algorithm based hyper-heuristic framework for the set packing problem

In recent years, hyper-heuristics have received massive attention from the research community as an alternative of meta-heuristics. In a hyper-heuristic, generation or selection of an effective heuristic among a pool of heuristics is an important and challenging task in the search process. At each iteration, a suitable heuristic can take the search process toward the global optimal solution. Moreover, some additional factors such as quality and the number of heuristics also affect the performance. In this paper, we propose an evolutionary algorithm based hyper-heuristic framework that incorporates dynamic selection of parameters. To test its generality, effectiveness and robustness, we apply this approach on two different NP-hard problems - set packing problem (SPP) and minimum weight dominating set (MWDS) problem. The proposed approach for the SPP and the MWDS problem has been evaluated respectively on their respective set of benchmark instances. Computational results show that the proposed approach for the SPP and MWDS problem perform much better than their respective state-of-the-art approaches in terms of the solution quality and computational time.

A dual ascent heuristic for obtaining a lower bound of the generalized set partitioning problem with convexity constraints

In this paper we propose a dual ascent heuristic for solving the linear relaxation of the generalized set partitioning problem with convexity constraints, which often models the master problem of a column generation approach. The generalized set partitioning problem contains at the same time set covering, set packing and set partitioning constraints. The proposed dual ascent heuristic is based on a reformulation and it uses Lagrangian relaxation and subgradient method. It is inspired by the dual ascent procedure already proposed in literature, but it is able to deal with right hand side greater than one, together with under and over coverage. To prove its validity, it has been applied to the minimum sum coloring problem, the multi-activity tour scheduling problem, and some newly generated instances. The reported computational results show the effectiveness of the proposed method.

An exact solution method for the capacitated item-sharing and crowdshipping problem

The item-sharing and crowdshipping problem combines two concepts of the sharing economy, namely item-sharing and crowdshipping. Item-sharing is about renting items among members of a sharing community. Crowdshipping addresses the transportation of these items through private people on trips they make anyway. The considered problem is to decide (1.) which request for an item to fulfill through which of the supplied items and (2.) who is doing the transport of rented items from the supply-locations to the request-locations. We generalize this problem with regard to crowdshippers’ capacity, meaning that each crowdshipper can transport a given number of items along his/her intended route. This results in a detour routing problem, where crowdshippers are routed through intermediate locations on the way from their actual origin location to their intended destination. We propose an exact solution method based on a set packing formulation for which a label setting procedure generates feasible crowdshipper routes a priori. We also describe how to derive a heuristic from the exact approach. Our experiments identify to what extend higher capacities of crowdshippers lead to more profitable routes and under which conditions a heuristic reduction of the method is required to cope with the complexity of the problem. We also show that the new exact method clearly outperforms procedures that were developed earlier for a setting where each crowdshipper can transport at most one single item.

Improved polyhedral descriptions and exact procedures for a broad class of uncapacitated p-hub median problems

This work focuses on a broad class of uncapacitated p-hub median problems that includes non-stop services and setup costs for the network structures. In order to capture both the single and the multiple allocation patterns as well as any intermediate case of interest, we consider the so-called r-allocation pattern with r denoting the maximum number of hubs a terminal can be allocated to. We start by revisiting an optimization model recently proposed for the problem. For that model, we introduce several families of valid inequalities as well as optimality cuts. Moreover, we consider a relaxation of the model that contains several sets of set packing constraints. This motivates a polyhedral study that we perform and that leads to the identification of many families of facets and other valid inequalities to the relaxed problem that, in turn, provide valid inequalities for the original model. Some of these families are too large for being handled directly. For those cases, separation algorithms are also presented. Finally, we gather all the above elements in a branch-and-cut procedure that we devise and implement for tackling the problem. The methodological developments proposed are tested computationally using data generated from the well-known AP data set.

On transversal and 2-packing numbers in uniform linear systems

A linear system is a pair (P,L) where L is a family of subsets on a ground finite set P, such that |l∩l′|≤1, for every l,l′∈L. The elements of P and L are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point. A subset T of points of P is a transversal of (P,L) if T intersects any line, and the transversal number, τ(P,L), is the minimum order of a transversal. On the other hand, a 2-packing set of a linear system (P,L) is a set R of lines, such that any three of them have a common point, then the 2-packing number of (P,L), ν2(P,L), is the size of a maximum 2-packing set. It is known that the transversal number τ(P,L) is bounded above by a quadratic function of ν2(P,L). An open problem is to characterize the families of linear systems which satisfies τ(P,L)≤λν2(P,L), for some λ≥1. In this paper, we give an infinite family of linear systems (P,L) which satisfies τ(P,L)=ν2(P,L) with smallest possible cardinality of L, as well as some properties of r-uniform intersecting linear systems (P,L), such that τ(P,L)=ν2(P,L)=r. Moreover, we state a characterization of 4-uniform intersecting linear systems (P,L) with τ(P,L)=ν2(P,L)=4.

On a family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation

Abstract In 1995 T. Matsui considered a special family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation. In 2012 the author demonstrated that all polytopes of this family appear as faces of polytopes associated with the following NP-complete problems: the travelling salesman problem, the 3-satisfiability problem, the knapsack problem, the set covering problem, the partial ordering problem, the cube subgraph problem, and some others. Here it is shown that none of the polytopes of the aforementioned special family (with the exception of the one-dimensional segment) can appear as a face in a polytope associated with the problem of the maximum independent set, the set packing problem, the set partitioning problem, and the problem of 3-assignments.

The rank pricing problem: Models and branch-and-cut algorithms

One of the main concerns in management and economic planning is to sell the right product to the right customer for the right price. Companies in retail and manufacturing employ pricing strategies to maximize their revenues. The Rank Pricing Problem considers a unit-demand model with unlimited supply and uniform budgets in which customers have a rank-buying behavior. Under these assumptions, the problem is first analyzed from the perspective of bilevel pricing models and formulated as a non linear bilevel program with multiple independent followers. We also present a direct non linear single level formulation bearing in mind the aim of the problem. Two different linearizations of the models are carried out and two families of valid inequalities are obtained which, embedded in the formulations by implementing a branch-and-cut algorithm, allow us to tighten the upper bound given by the linear relaxation of the models. We also study the polyhedral structure of the models, taking advantage of the fact that a subset of their constraints constitutes a special case of the Set Packing Problem, and characterize all the clique facets. Besides, we develop a preprocessing procedure to reduce the size of the instances. Finally, we show the efficiency of the formulations, the branch-and-cut algorithms and the preprocessing through extensive computational experiments.

An Efficient Mixed Integer Linear Programming Model for the Minimum Spanning Tree Problem

Finding a minimum spanning tree in a given network is a famous combinatorial optimization problem that appears in different engineering applications. Even though this problem is solvable in polynomial time, having efficient mathematical programming models is important as they can provide insights for formulating larger models that integrate other decisions in more complex applications. In the literature, there are ten different integer and mixed integer linear programming (MILP) models for this problem. They are variants of set packing, cuts, network flow and node level formulations. In addition, this paper introduces an efficient node level MILP model. Comparisons for the eleven models are provided. First, the models are compared in terms of the number of decision variables and the number of constraints. Then, computational comparisons using a commercial MILP solver on sets of randomly generated instances of different sizes are conducted. Results provide evidence that the proposed MILP model is competitive in terms of the computational time needed for proving optimality of generated solutions for instances with up to 50 nodes. Meanwhile, the LP relaxation of a multi-commodity flow MILP model which has integer polyhedron provides stable computational times despite its larger size.

The strength of Dantzig–Wolfe reformulations for the stable set and related problems

Dantzig–Wolfe reformulation of an integer program convexifies a subset of the constraints, which yields an extended formulation with a potentially stronger linear programming (LP) relaxation. We would like to better understand the strength of such reformulations in general. As a first step we investigate the classical edge formulation for the stable set problem. We characterize weakest possible Dantzig–Wolfe reformulations (with LP relaxations not stronger than the edge formulation) and strongest possible reformulations (yielding the integer hull). We (partially) extend our findings to related problems such as the matching problem and the set packing problem. These are the first non-trivial general results about the strength of relaxations obtained from decomposition methods, after Geoffrion’s seminal 1974 paper about Lagrangian relaxation.