Set Packing Research Articles

Set Packing Optimization by Evolutionary Algorithms with Theoretical Guarantees.

The set packing problem is a core NP-complete combinatorial optimization problem which aims to find the maximum collection of disjoint sets from a given collection of sets, S, over a ground set, U. Evolutionary algorithms (EAs) have been widely used as general-purpose global optimization methods and have shown promising performance for the set packing problem. While most previous studies are mainly based on experimentation, there is little theoretical investigation available in this area. In this study, we analyze the approximation performance of simplified versions of EAs, specifically the (1+1) EA, for the set packing problem from a theoretical perspective. Our analysis demonstrates that the (1+1) EA can provide an approximation guarantee in solving the k-set packing problem. Additionally, we construct a problem instance and prove that the (1+1) EA beats the local search algorithm on this specific instance. This proof reveals that evolutionary algorithms can have theoretical guarantees for solving NP-hard optimization problems.

On the Parameterized Complexity of Compact Set Packing

The Set Packing problem is, given a collection of sets S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}$$\\end{document} over a ground set U, to find a maximum collection of sets that are pairwise disjoint. The problem is among the most fundamental NP-hard optimization problems that have been studied extensively in various computational regimes. The focus of this work is on parameterized complexity, Parameterized Set Packing (PSP): Given parameter r∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\in {\\mathbb N}$$\\end{document}, is there a collection S′⊆S:|S′|=r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal {S}' \\subseteq \\mathcal {S}: |\\mathcal {S}'| = r$$\\end{document} such that the sets in S′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}'$$\\end{document} are pairwise disjoint? Unfortunately, the problem is not fixed parameter tractable unless W[1]=FPT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {W[1]} = \ extsf {FPT} $$\\end{document}, and, in fact, an “enumerative” running time of |S|Ω(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\mathcal {S}|^{\\Omega (r)}$$\\end{document} is required unless the exponential time hypothesis (ETH) fails. This paper is a quest for tractable instances of Set Packing from parameterized complexity perspectives. We say that the input (U,S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({U},\\mathcal {S})$$\\end{document} is “compact” if |U|=f(r)·poly(log|S|)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|{U}| = f(r)\\cdot \ extsf {poly} ( \\log |\\mathcal {S}|)$$\\end{document}, for some f(r)≥r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(r) \\ge r$$\\end{document}. In the Compact PSP problem, we are given a compact instance of PSP. In this direction, we present a “dichotomy” result of PSP: When |U|=f(r)·o(log|S|)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|{U}| = f(r)\\cdot o(\\log |\\mathcal {S}|)$$\\end{document}, PSP is in FPT, while for |U|=r·Θ(log(|S|))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|{U}| = r\\cdot \\Theta (\\log (|\\mathcal {S}|))$$\\end{document}, the problem is W[1]-hard; moreover, assuming ETH, Compact PSP does not admit |S|o(r/logr)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\mathcal {S}|^{o(r/\\log r)}$$\\end{document} time algorithm even when |U|=r·Θ(log(|S|))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|{U}| = r\\cdot \\Theta (\\log (|\\mathcal {S}|))$$\\end{document}. Although certain results in the literature imply hardness of compact versions of related problems such as Setr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r$$\\end{document}-Covering and Exactr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r$$\\end{document}-Covering, these constructions fail to extend to Compact PSP. A novel contribution of our work is the identification and construction of a gadget, which we call Compatible Intersecting Set System pair, that is crucial in obtaining the hardness result for Compact PSP. Finally, our framework can be extended to obtain improved running time lower bounds for Compactr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r$$\\end{document}-VectorSum.

An efficient hybrid adaptive large neighborhood search method for the capacitated team orienteering problem

This paper focuses on a NP-Hard vehicle routing problem with profits, and presents a hybrid heuristic solution approach for the Capacitated Team Orienteering Problem (CTOP), which is a variant of the well-known team orienteering problem. In the CTOP, given a set of homogeneous fleet and a set of clients to each is associated a deterministic profit and volume demand, one maximizes the collected profits for serving customers by operating a set of routes while respecting vehicles’ limited time and volume capacities. Our proposed method to solve the CTOP is a Hybrid Adaptive Large Neighborhood Search (HALNS) heuristic. Several removal and insertions operators with different node selection strategies and local search procedures are developed to generate solutions in the first step. In the second step, solutions’ routes are passed to a set packing problem which is solved using an exact branch-and-cut procedure in order to obtain the best solution. Compared to all the state-of-the-art methods, extensive computational experiments show that our method outperforms in terms of solution quality and/or computational times when tested on the small and large-scale benchmark CTOP instances respectively proposed by Archetti et al. (2009) and Tarantilis et al. (2023). Furthermore, our hybrid method improves 10 of the best known solutions for large-scale benchmark instances.

An attention model for the formation of collectives in real-world domains

We consider the problem of forming collectives of agents inherent in application domains aligned with Sustainable Development Goals 4 and 11 (i.e., team formation and ridesharing, respectively). We propose a general solution approach based on a novel combination of an attention model and an integer linear program (ILP). In more detail, we propose an attention encoder-decoder model that transforms a collective formation instance to a weighted set packing problem, which is then solved by an ILP. Results on collective formation problems inherent in the ridesharing and team formation domains show that our approach provides comparable solutions (in terms of quality) to the ones produced by state-of-the-art approaches specific to each domain. Moreover, our solution outperforms the most recent general approach for forming collectives based on Monte Carlo tree search.

Algorithms and computational study on a transportation system integrating public transit and ridesharing of personal vehicles

The potential of integrating public transit with ridesharing includes shorter travel time for commuters and higher occupancy rate of personal vehicles and public transit ridership. In this paper, we describe a centralized transit system that integrates public transit and ridesharing to reduce travel time for commuters. In the system, a set of ridesharing providers (drivers) and a set of public transit riders are received. The optimization goal of the system is to assign riders to drivers by arranging public transit and ridesharing combined routes for as many riders as possible subject to shorter commuting time. As an exact algorithm approach, we give a (0,1) integer linear programming (ILP) formulation based on a hypergraph representation of the problem. Leveraging the ILP and hypergraph, we give approximation algorithms based on LP-rounding and hypergraph matching/weighted set packing, respectively. As a case study, we conduct an extensive computational study based on real-world public transit dataset and ridesharing dataset in Chicago city. To evaluate the effectiveness of the transit system and our algorithms, we generate data instances from the datasets. The experimental results show that more than 60% of riders are assigned to drivers on average, riders’ commuting time is reduced by 23% and vehicle occupancy rate is improved to almost 3. Our proposed algorithms are efficient for practical scenarios.

Maximizing value yield in wood industry through flexible sawing and product grading based on wane and log shape

The optimization of sawing processes in the wood industry is critical for maximizing efficiency and profitability. The introduction of computerized tomography scanners provides sawmill operators with three-dimensional internal models of logs, which can be used to assess value and yield more accurately. We present a methodology for solving the sawing optimization problem employing a flexible sawing scheme that allows greater flexibility in cutting logs into products while considering product quality classes influenced by wane defects. The methodology has two phases: preprocessing and optimization. In the preprocessing phase, two alternative algorithms are given that generate and evaluate the potential sawing positions of products by considering the 3D surface of the log, product size requirements, and product quality classes. In the optimization phase, a maximum set-packing problem is solved for the preprocessed data using mixed-integer programming (MIP), aiming to obtain a feasible cut pattern that maximizes value yield. This is implemented in a system named FlexSaw, which takes advantage of parallel computation during the preprocessing phase and utilizes a MIP solver during the optimization phase. The proposed sawing methods are evaluated on the Swedish Pine Stem Bank. Additionally, FlexSaw is compared with an existing tool that utilizes cant sawing. Results demonstrate the superiority of flexible sawing. While the practical feasibility of implementing a flexible way of sawing logs is constrained by the limitations of current sawmill machinery, the potential increase in yield promotes the exploration of alternative machinery in the wood industry.

Two‐stage stochastic one‐to‐many driver matching for ridesharing

AbstractWe introduce a modeling framework for stochastic rider‐driver matching in many‐to‐one ridesharing systems, in which drivers have to be selected before the exact rider demand is known. The modeling framework allows for the use of driver booking fees and penalties for unmatched drivers, therefore supporting different system operating modes. We model this problem as a two‐stage stochastic set packing problem. To tackle the intractability of the stochastic problem, we introduce three model approximations and evaluate them on a large set of benchmark instances for three different system operating modes. Our computational experiments show the superiority of some model approximations over others and provide valuable insights on the impact of penalties and booking fees on the system's profitability and user satisfaction.

How heavy independent sets help to find arborescences with many leaves in DAGs

Trees with many leaves have applications for broadcasting a message to all recipients simultaneously. Internal nodes of a broadcasting tree require more expensive technology to forward the messages received. We address a problem that captures the main goal: finding spanning trees with few internal nodes in a given network. The Maximum Leaf Spanning Arborescence problem consists of, given a directed graph D, finding a spanning arborescence of D, if one exists, with the maximum number of leaves. This problem is NP-hard in general and MaxSNP-hard in the rooted directed acyclic graphs class. This paper explores a relationship between Maximum Leaf Spanning Arborescence in rooted directed acyclic graphs and maximum weight set packing. The latter problem is related to independent sets on particular classes of intersection graphs. Exploiting this relation, we derive a 7/5-approximation for Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs, improving on the previous 3/2-approximation.

The lexicographic method for the threshold cover problem

Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size k if its edges can be covered using k threshold graphs. Chvátal and Hammer, in 1977, defined the threshold dimensionth(G) of a graph G to be the least integer k such that G has a threshold cover of size k and observed that th(G)≥χ(G⁎), where G⁎ is a suitably constructed auxiliary graph. Raschle and Simon (1995) [9] proved that th(G)=χ(G⁎) whenever G⁎ is bipartite. We show how the lexicographic method of Hell and Huang can be used to obtain a completely new and, we believe, simpler proof for this result. For the case when G is a split graph, our method yields a proof that is much shorter than the ones known in the literature. Our methods give rise to a simple and straightforward algorithm to generate a 2-threshold cover of an input graph, if one exists.

SG-APSIC1084: Reduce cost and resterilization rate of reusable medical device sets by reorganizing and rearranging packaging and process

Objectives: We evaluated the resterilization rate, user satisfaction, and cost of resterilization after rearranging and packing of reusable instrument sets. Methods: For 1 month in July 2018, we conducted an observational prospective study in 39 service departments for which sterilization and instrument packing was done by the central sterile supply department (CSSD). Common sterile instrument sets (eg, intercostal drainage (ICD) sets, bone-marrow aspiration sets, or suture sets) were analyzed to set up basic surgical instruments for common procedures and specific instruments for each procedure. Sets for common procedures were then packed and rearranged for use universally in various procedures separately from specific instruments. A questionnaire survey was delivered to all 39 service departments to evaluate user satisfaction. The resterilization rates and cost analyses before and after the rearranging and packing were compared for their effectiveness. The data were analyzed using descriptive statistics for percentage, mean, standard deviation, and inferential statistics. Categorical data were analyzed using the χ2 test and continuous data were analyzed using a t test with significance level of 0.05. Results: The resterilization rate decreased significantly from 7.1% to 0.1%. The cost of resterilization decreased from 76,500 Thai baht (US $2,287) to 4,800 Thai baht (US $143) within 1 month. Overall, user satisfaction regarding this intervention was 85.2%. Conclusions: This study highlights the need for the evaluation of process and customer demand to improve user satisfaction and reduce hospital cost by customizing the sterilization packaging and rearranging process.

Constraint relaxation for the discrete ordered median problem

This paper compares different exact approaches to solve the Discrete Ordered Median Problem (DOMP). In recent years, DOMP has been formulated using set packing constraints giving rise to one of its most promising formulations. The use of this family of constraints, known as strong order constraints (SOC), has been validated in the literature by its theoretical properties and because their linear relaxation provides very good lower bounds. Furthermore, embedded in branch-and-cut or branch-price-and-cut procedures as valid inequalities, they allow one to improve computational aspects of solution methods such as CPU time and use of memory. In spite of that, the above mentioned formulations require to include another family of order constraints, e.g., the weak order constraints (WOC), which leads to coefficient matrices with elements other than {0,1}. In this work, we develop a new approach that does not consider extra families of order constraints and furthermore relaxes SOC -in a branch-and-cut procedure that does not start with a complete formulation- to add them iteratively using row generation techniques to certify feasibility and optimality. Exhaustive computational experiments show that it is advisable to use row generation techniques in order to only consider {0,1}-coefficient matrices modeling the DOMP. Moreover, we test how to exploit the problem structure. Implementing an efficient separation of SOC using callbacks improves the solution performance. This allows us to deal with bigger instances than using fixed cuts/constraints pools automatically added by the solver in the branch-and-cut for SOC, concerning both the formulation based on WOC and the row generation procedure.

New variants of the simple plant location problem and applications

This work revisits novel variants of the Simple Plant Location Problem (SPLP), which were originally presented in the Ph.D. thesis Set packing, location and related problems (2019). They consist in considering different requirements on clients allocation to facilities, namely incompatibilities or co-location for some specific pairs of clients. These correspond to “cannot-link” and “must-link” constraints previously studied in the related topic of clustering. Set packing formulations of these new location problems are presented, together with families of valid inequalities and facets. Computational experience supporting the effectiveness of the resulting cuts into a branch and cut procedure is exposed. The comprehensive revision on these new variants of the SPLP is completed by illustrative applications, including the design of telecommunication networks and interesting connections in the field of Genetics, and also by the analysis of open questions and future research directions.

A fast approximation algorithm for the maximum 2-packing set problem on planar graphs

A fast approximation algorithm for the maximum 2-packing set problem on planar graphs

Dynamic Kernels for Hitting Sets and Set Packing

Computing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d, and a parameter k. The task is to compute a new hypergraph, called a kernel, whose size is polynomial with respect to the parameter k and which has a size-k hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size k^d (which is a polynomial as d is a constant), and they do so in time mcdot 2^d {text {poly}}(d) for a small polynomial {text {poly}}(d) (which is linear in the hypergraph size for d fixed). We generalize this task to the dynamic setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-k^d kernel. This paper presents a deterministic solution with worst-case time 3^d {text {poly}}(d) for updating the kernel upon inserts and time 5^d {text {poly}}(d) for updates upon deletions. These bounds nearly match the time 2^d {text {poly}}(d) needed by the best static algorithm per hyperedge. Let us stress that for constant d our algorithm maintains a hitting set kernel with constant, deterministic, worst-case update time that is independent of n, m, and the parameter k. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-k hitting sets in d-hypergraphs with update times O(1) and query times O(c^k) where c = d - 1 + O(1/d) equals the best base known for the static setting.

Packings of partial difference sets

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group \(G\). This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals important structural information about the packing. Identifying this subgroup allows us to formulate a recursive lifting construction of packings in abelian groups of increasing exponent, as well as a product construction yielding packings in the direct product of the starting groups. We also study packings of certain negative Latin square type partial difference sets of maximum possible size in abelian groups, all but one of which have identical parameters, and show how to produce such collections using packings of Latin square type partial difference sets.Keywords: Finite abelian group, packing, partial difference set.Mathematics Subject Classifications: 05B10, 20K01

Learning generalized strong branching for set covering, set packing, and 0–1 knapsack problems

Branching on a set of variables, rather than on a single variable, can give tighter bounds at the child nodes and can result in smaller search trees. However, selecting a good set of variables to branch on is even more challenging than selecting a good single variable to branch on. Generalized strong branching extends the strong branching concepts developed for choosing a single variable to choosing a set of variables. As the computational requirements of a full implementation of strong branching are prohibitive, we use extreme gradient boosting to train a model to predict the ranking of (sets of) candidate variables. An extensive computational study using instances from three well-known classes of optimization problems demonstrates that branching on sets of variables outperforms branching on a single variable, that a learned model can be used effectively to select among (sets of) candidate variables, and that the learned strong branching strategies outperform the default branching strategy of state-of-the-art commercial solver CPLEX in terms of both the number of nodes explored in the search tree and the time it takes to explore the search tree.

On the complexity of surrogate and group relaxation for integer linear programs

Surrogate and group relaxation have been used to compute bounds for various integer linear programming problems. We prove that (a) when only inequalities are surrogated, the surrogate dual is NP-hard, but solvable in pseudo-polynomial time under certain conditions; (b) when equations are surrogated, the surrogate dual exhibits unusual complexity behaviour; (c) the group relaxation is NP-hard for the integer and 0-1 knapsack problems, and strongly NP-hard for the set packing problem.

Representative families for matroid intersections, with applications to location, packing, and covering problems

We show algorithms for computing representative families for matroid intersections and use them in fixed-parameter algorithms for set packing, set covering, and facility location problems with multiple matroid constraints. We complement our tractability results by hardness results.

On dominating set polyhedra of circular interval graphs

Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. (2008) take advantage of this relationship to propose a complete linear description of the stable set polytope on circular interval graphs. In this paper we follow similar ideas to obtain a complete description of the dominating set polytope on the same class of graphs. As in the packing case, our results are established for a larger class of covering polyhedra of the form Q∗(A,b)≔convx∈Z+n:Ax≥b, with A a circular matrix and b an integer vector. These results also provide linear descriptions of polyhedra associated with several variants of the dominating set problem on circular interval graphs.

Solving the Set Packing Problem via a Maximum Weighted Independent Set Heuristic

The set packing problem (SPP) is a significant NP-hard combinatorial optimization problem with extensive applications. In this paper, we encode the set packing problem as the maximum weighted independent set (MWIS) problem and solve the encoded problem with an efficient algorithm designed to the MWIS problem. We compare the independent set-based method with the state-of-the-art algorithms for the set packing problem on the 64 standard benchmark instances. The experimental results show that the independent set-based method is superior to the existing algorithms in terms of the quality of the solutions and running time obtained the solutions.